Space Solar Power For Launch Loop
Work in progress! _Most of these numbers have not been double checked._
The space solar power community offers numbers like 7 kg per kilowatt in orbit for space solar power, solar energy converted to <10 GHz microwaves and beamed to Earth. This is built on a chain of optimistic assumptions, and assumes launch costs much cheaper than the current $20,000/kg to GEO, but still using rocket thrust. Too many oughta works is a not oughta work.
A launch loop can launch 5000 kg payloads into GTO (Geosynchronous Transfer Orbit) vastly cheaper than any groundlaunched rocket. Low thrust propellantthrifty electric engines (such as VASIMR) can raise perigee and circularize the orbit of a geosynchronous power satellite, which can be used to feed more power and increase the launch rate of a launch loop.
Starting with a 500 MW thermal power plant, perhaps 300 MW will be needed for deflection magnets and drag losses, leaving 200 MW to power launches. A 6 tonne sled plus the vehicle launched into GTO might result in 2 tonnes of SSPS mass delivered to GEO after a sequence of orbit raising manuevers and construction assembly steps.
5° Inclination
The first "low mass" launch loops will be perhaps 5° south of the equator, to avoid January storms. This means they will launch into 5° inclined orbits. At first blush, this is a problem, since a geosynchronous SSPS is sometimes 5 degrees farther south in the sky, limiting the northern footprint. However, some inclination is good, so that the huge transmit energy of the SSPS does not blind receiver dishes pointed at communication satellites in equatorial GEO orbits. This will require significant angular spacing above or below the crowded GEO orbit. SSPS satellites will probably need to shut down or reduce power as their orbits pass through the equatorial plane. It is better if they pass through the equatorial plane quickly, perhaps using that time for brief maintenance, calibration, and generation of stationkeeping thrust.
The 5° inclination means some fraction of the satellites cannot reach as far to the north, while others can reach farther. If they switch between northerly and southerly ground rectennas (pulsed power helps with this) as they cycle north to south in a 24 hour cycle, the whole constellation of SSPS satellites can provide complete coverage farther to the north, while still providing maximum overlapping coverage for equatorial customers (where most of the world's population lives).
So ... no inclination change to equatorial, unless the destination orbit is for a comsat feeding fixeddirection ground dishes. Those comsats will be important, but not a large fraction of the total launch market.
SSPS launch, construction, and deployment
Highly elliptical orbits are relatively cheap with the launch loop; indeed, they can be cheaper than low orbits, because raising perigee from 100 km loop altitude to a low drag 500 km perigee requires less delta V at apogee. Here is a sequence of steps to launch, assemble, and deploy an SSPS to a 5° inclined circular geosynchronous orbit:
period 
radius 
altitude 
v apogee 
delta V from below 

hours 
km 
km 
m/s 
m/s 
stage 

1164.9 

10.485 
6478.0 
100.0 
1586.8 
9855.9 

10.615 
6878.0 
500.0 
1628.4 
41.6 

11.967 
10959.4 
4581.4 
1975.0 
346.6 

23.934 
42164.2 
35786.2 
3074.7 
1099.7 


268.2 
Elevator and Loop Launch
The launch loop is a motor, and motor losses are directly proportional to force, and (if properly designed) a very weak function of velocity. The acoustic elevator drives vehicles up through the thick atmosphere, increasing in speed until the vehicles are moving faster than Mach 2. For the last 10 kilometers of climb before west station at 70 km, they are actively slowing down, "eyeballs out" added to gravity, perhaps 20 m/s² from 600+ m/s and putting power back into the elevator tether, adding to the power that lifts the next climbing vehicle. If the acoustic elevator is 80% efficient (WAG, for both lift and generation), and 30% of lift power is dissipated against air drag (WAG), then the endtoend energy efficiency will be approximately 50%.
The "delta V" energy from ground to 70 km (at 100% efficiency) is ½v² = 0.5×1165² J/kg, or 0.7 MJ/kg. With 50% inefficiency, 1.4 MJ/kg. Call it 2 MJ/kg.
The launch motor is very efficient, perhaps 98%, and from 70 km altitude (at relatively low speed, at the west/front end of the loop) to 100 km altitude (at the orbital speed eastern end of the loop) the vertical change is small (perhaps 0.3 MJ/kg) and the added kinetic energy quite large; ½v² = 0.5×9860² = or about 49 MJ/kg. Assuming 98% efficiency and adding the elevator energy, the launch loop system input energy per kilogram launched is around 52 MJ/kg, or 14.5 kWh/kg . That kilogram includes the launch sled and first kick motor and propellant needed for later stages of the process, as well as the disposables, support robots, replacement parts, etc. associated with assembly, but it is much less than the energy needed for a surfacetoorbit staged rocket.
Assume the launch sled with nose cone weighs 1000 kg added to a 5000 kg vehicle, and that it is decelerated with 50% energy recovery and completely recycled. That means the energy used so far increases by 10%, to about 16 kWh/kg.
With 200 MW available for launch, we can launch about 12 tonnes per hour.
Rapid Thrust to Safe Perigee
The vehicle (minus the sled) is now in a highly elliptical orbit, with a perigee at 100 km altitude and an apogee at 42164 km GEO radius, 35786 km altitude . It will reenter without additional velocity added at apogee to raise the perigee to 500 km. That requires a delta V of 42 meters per second; if we use a small solid rocket with an I_{SP} of 200 seconds (the shuttle solids were 250 seconds), which means an exhaust velocity of 2000 meters per second, the rocket must expel about 0.021 kg of propellant per kilogram of vehicle to accomplish this boost. Considering only the energy cost of the propellant, that is 42 kJ per kilogram of vehicle, but let's assume that rocket making is very inefficient and round that way up to 720 kJ/kg, or 0.2 kWh/kg. This does not add much to the 16 kWh/kg energy total, but the weight of the motor subtracts from the mass, as does the chance of mishap (solid kick motor doesn't fire, or explodes). As a WAG, round that up to net energy of 17 kWh/kg for net mass delivered to a 500 × 35786 km elliptical orbit. At this altitude, drag is small, so we can deploy large solar panels for the next stage.
Slow Thrust to 12 hour Construction Orbit
With low drag and plenty of time for this next stage, we will raise perigee to an altitude of 4580 km, resulting in a "12 hour" orbit, a power satellite construction orbit with a perigee that repeats over the same portion of the Earth once per solar day (and on the opposite side of the Earth, ditto). In addition, we will choose the orbital epoch (timing) so that the entire inclined orbit is in full sunlight. IMHO, we will construct space solar power satellites with 100% robotic teleoperation, the timing and location will mean that occasional human visitors suffering an accident can be deorbited to a hospital relatively quickly (perhaps a day), and that operations will always be in full sunlight.
But we aren't to that orbit yet; for moving mass most efficiently we will use high ISP '''VASIMR''' electric engines with argon propellant. The vehicle we just launched will not have its own VASIMR engine and propellant; instead, there will be a fleet of VASIMR orbital tugs that cycle between the 500 km perigee "safe" orbit and the 4580 km perigee "construction" orbit, slowly adding 360 m/s at apogee to accelerate 4900 kg vehicles (minus the solid stage propellant weight), and subtracting 360 m/s from the unladen tug to return to the lower orbit and pick up another load.
Presume that most of the weight of the orbital tug is solar panels and propellant in approximately equal proportions, and that the solar panels (hardened for frequent trips through the van Allen belt) mass 20 kg/kW. Presume an "thrust time" of 50% of the orbit, so the energy contributed by the panels is the half the total orbit raising time T (in hours), or 25Whr/kg of fuel plus solar panel. This energy is used to eject the propellant and create delta V. For the first approximation, we will consider only the climb (laden with vehicle payload), not the unladen descent.
The faster we push the fuel, the more I_{SP} we get, up to perhaps 5000 seconds (50 km/s exhaust), with 70% thruster efficiency. For 360 m/s delta V, that is a mimimum propellant fraction of 360/50000, or 0.72%; round that up to 1% for a sloppy accomodation of thruster efficiency, 50 kg of argon propellant with a total energy of 0.5 × 50 × 50000² or 62.5 GJ for a 5000 kg vehicle, 12.5 MW per vehicle kg. We are providing that with a 50 kg, 1.25 kW (average) solar panel, so this transfer will require 5e7 seconds, or 580 days.
Oopsie. With 20%/year highrisk capital cost, that increases system cost by almost 35%. Lets use more propellant, more solar panel, and lower ISP. What does 1000 second ISP do for us ... 250 kg of propellant, 250 kg of solar cell?
Now we need 0.5 × 250 x 10000² or 12.5 GJ from a 6.25 kW (average) solar panel, 2e6 seconds or 23 days. At 20%/year capital cost, that increases system cost by 1.3% . Except that we also need to ship up the 250 kg of argon for our orbital tug (presumably a second coordinated rendezvous with the tug), which increases system cost by another (250/5000) or 5%, perhaps 8% considering tankage and boiloff and refrigeration.
The answer will probably be somewhere in between, and we will let the scheduling engineers figure this out. My wild guess is that this tug will cost us about 10% in "energy cost".
SSPS Construction
Anyone who has ever been to a factory or a construction site knows that construction is less than 100% mass efficient. Your house is made of wood and concrete ... and windows and pipes and wire and fasteners, etc. It is also made with trucks and cranes and backhoes and many workers with tools, which are partly reusable, but wear out or break over time. We've built houses for thousands of years, but we still haul off dumpsters full of trimmings, broken tools, defective components, and miscellaneous scrap. In spite of the best efforts of clever engineers, I cannot imagine SSPS construction will ever be scrap free. Since nobody has ever built an SSPS, we can presume the first ones will always fail (100% scrap), the subsequent ones will have perhaps 50% scrap rates, perhaps 20% scrap rates in a superbly competent future. This suggests making the first ones as small as practical, even if they are less efficient.
I am an SSPS maverick. Much ink and chatroom bandwidth has been spilt whinging about the potential weaponization of SSPS. I worry about interference. One way around this is 183 GHz SSPS, a very short wavelength that is absorbed by water vapor, completely blocked by moisture in the troposphere before it reaches the ground. It can still heat up high flying airplanes, but not cities or people, or even telescopes on high mountains. It could be detectable through bone dry arctic air, but the poles are below the horizon for SSPS.
The rectennas will be on aerostats (shaped balloons) in the lower stratosphere (above the fastest winds), connected with power cables to the ground. It is very cold in the lower stratosphere, and even in the thin air, the wind can carry off a lot of heat. Without blooming, or health risks, we can run power levels of perhaps 20 kW/m².
The voltage per meter on a dipole "rectenna" element will be small, and diode drop losses high. But why use a dipole and a diode, to make a small voltage. Magnetrons use high voltage and a magnetic field to make microwaves, fairly efficiently. I'm just guessing here, but can we reverse this process, use microwaves and a guided channel to push electrons up to high voltage? There may be a problem with "entendue"  or this may be a way to make a small element produce high voltage directly.
Or not. In any case, if we pulse the SSPS power, we can average to 20 kW/m² with 1MW/m² pulses and 2% duty cycle. That gives us 27 kV per meter on our 0.8 mm halfwave dipoles, or 22 volts; a 1 volt diode drop wouldn't hurt much.
A major advantage of 183 GHz and 1.6 mm wavelength is that all the optics scale proportionally. For a 40,000 kilometer distance from SSPS satellite to high latitude rectenna, the beam size from a 400 meter orbiting transmitter is 200 meters, 40,000 km away; If the receiver is a disk powered at 20 kW/m², that is 630 MW. Assuming 65% conversion efficiency, that is about 400 MW delivered to the terrestrial customer. Asssuming 50% conversion efficiency at the transmitter end, and 20 kg/kW, that is a 13,000 tonne power satellite, for which we need to launch about 30,000 tonnes of mass from the launch loop on the ground.
With 200 MW launch power netting about 10 tonnes per hour to the construction area, that is 3000 hours per new SSPS, about 1.2 GW of new generation per year. This is not the big numbers that space solar power advocates demand right now only, but it will have spectacular effects feeding a launch loop, and later a large array of them.
Slow Thrust to GEO delivery
This is a power satellite in a highly elliptical orbit, we need to add another 1100 m/s of delta V to put it in geosynchronous orbit. Again, we will use VASIMR tugs (many of them) powered by the entire 1.2 GW array solar panel. If we assume 50000 m/s exhaust velocity and 70% thrust efficiency, then 1.2 GW will accelerate 1 kg of argon per second and create 3.5e4 newton seconds of thrust, accelerating the 1.3e7 kg SSPS at 0.0027 m/s², and achieving 1100 m/s of delta V in 410,000 seconds, less 5 days, and expending 410 tonnes of argon.
We will leave the engines and some tanked argon attached, so we can quickly return the SSPS to the construction orbit for rework. Which will be often, until the system is debugged.
Implications for Launch Rates
It may take a year to get the first SSPS debugged and feeding power ... to rectennas connected to the launch loop. With an extra 400 MW, launch rates triple. In six months, the amount of power feeding the loop can triple again. The first loop can use up to 6 GW of launch power, and put 400 tonnes into orbit per hour. We should build at least three, preferably ten, launch loops so we can take a few down for repair and upgrade, but in a couple of years we are putting perhaps 3000 tonnes into orbit per hour, 1500 net tonnes of SSPS in orbit per hour, perhaps 1000 times 400 MW of additional power delivered to earth per year from these first launch loops.
This is just the beginning.
A 3 kg/m rotor launching 5 tonne vehicles is about the minimum size launch loop we can build in our windy atmosphere, but there is no upper limit on rotor and vehicle mass. A 100 kg/m rotor system can launch 20 tonne "vehicles", which might be fully loaded 40 foot intermodal shipping containers plucked off a container ship, and launch 600 of them per hour. A 1000 kg/m rotor system could launch the entire cargo of 60 Panamax container ships per day, almost 3 million tonnes per hour. At 30,000 launched tonnes per 400 MW of ground power, that is 40 GW of SSPS launched per hour, 350 TW per year, far more energy than the world will ever need.
We will probably learn how to build stronger cables, and faster rotors, which will also increase launch rates. My hope is that we can double global ocean traffic, moving more material launched into space than the world trades now.
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