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 (CHECKME) of SSPS mass delivered to GEO after a sequence of orbit raising manuevers and construction assembly steps.
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 
apogee 
ΔV 
stage 
remaining 
rocket 
propl. 
time 
hours 
km 
km 
V m/s 
m/s 

kg 
km/s 
kg 
days 
0.7 MJ/kg equivalent ↔ 
1165 
5000 

0 


10.485 
6478 
100 
1587 
9856 
5000 

0 


10.615 
6878 
500 
1628 
42 
4917 
2.5,s 
83 
0.2 

11.967 
10959 
4581 
1975 
347 
4712 
10,v 
205 
9.7 


2260 


23.934 
42164 
35786 
3075 
1100 
2265 
20,v 
132 
5.8 


347 

10,v 
35 
24.4 


268 

20,v 





,v = vasimr, . . . ,s = solid
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 5000 kg 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 104 kg of propellant 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 use large solar panels for the next stage.
Slow Thrust to 12 hour Construction Orbit
With low drag and plenty of time for this next boost, 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 99% robotic teleoperation. The timing and location will mean that occasional short term 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 the construction orbit yet. To reduce propellant mass, and the risk of fire or explosion, 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 reusable VASIMR orbital tugs that cycle between the 500 km perigee "safe" orbit and the 4580 km perigee "construction" orbit, slowly adding 347 m/s at apogee to accelerate 4900 kg vehicles (minus the 80ksolid stage propellant weight), and subtracting 347 m/s from the unladen tug to return to the lower orbit and pick up another load.
Assume a 1000 kg orbital transfer tug, with a large and radiationhardened solar panel providing 50 kW of power (there will be many of these in mulitiple safe orbit constellations). Along with the tugs (which cycle up and down between safe and construction orbits), the safe orbits will have 5000 kg tanks of superchilled liquid/solid argon as fuel depots, launched separately to refill many tugs. The argon might launch at 60K and 1600 kg/m³, in a large insulated tank, with ullage space for boiloff. Towards the end of an SSPS construction cycle, more full argon tanks will be transported by tugs to the construction orbit, and used for deployment transfer of the complete SSPS to GEO.
While VASIMR engines can be run at very high ISP (this makes sense for interplanetary missions), for the acceleration to the construction orbit we will tune them for an ISP of 1000 seconds and an exhaust velocity of 10 km/s, more than double the best hydrogen/oxygen stages. This uses more propellant but less power, to make more thrust. Presume that most of the weight of the 200 kg orbital tug is 50 kW of radiationhardened solar panels. Presume a "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 climb will require almost 10 days and 20 to 21 orbits. Since payload is being delivered from many somewhat different safe orbits, we will adjust timing to converge on the same construction orbit. Later, as we increase launch loop power and throughput, we will operate in many constellations of safe orbits and construction orbits.
Scrap
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.
SSPS Construction
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 be used (in theory) for heating high flying airplanes (difficult in practice if the planes have reflective hulls), but not cities or people, or even telescopes on high mountains. 183 GHz millimeter waves 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
A 1.2 GW , 12,000 tonne power satellite in a highly elliptical orbit can deliver a prodigeous amount of power to high ISP (2000 seconds) VASIMR to add another 1100 m/s of delta V and put it in a circular geosynchronous orbit. Assuming 20000 m/s exhaust velocity and 70% thrust efficiency, then 1.2 GW will accelerate 4.2 kg of argon per second and create 8.4e4 newton seconds of thrust, accelerating the 1.2e7 kg SSPS at 0.007 m/s², with 50% average duty cycle (near apogee only), and achieve 1100 m/s of delta V in 310,000 seconds, about 4 days, while expending 1300 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.
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 in this inclned orbit will sometime be 5 degrees farther south in the sky, limiting the northern footprint. However, the SSPS should not be collinear with a much less powerful communication satelite, or the sidelobe noise associated with the huge transmit energy of the SSPS will blind receiver dishes on the ground pointed towards that satellite. Minimizing the energy not only requires incredible sidelobe attenuation, an SSPS must be far to the north or south of the equatorial plane so that the offaxis attenuation of typical asymmetric receive dishes is adequate. SSPS satellites probably must 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 the generation of stationkeeping thrust, while other SSPS satellites to the north (or south for tropical or southern customers) supply power instead.
The 5° inclination means the southerly fraction of the satellites cannot reach as far to the north, while northerly satellites can reach farther north. If the satellites 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 ... an inclination change from inclined to equatorial orbit is not necessary, unless the satellite is a comsat feeding fixeddirection ground dishes. Communication satellites will still be important, but a small fraction of the total launch market.
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 planet Earth will ever need, but needed to extending human presence into the solar system.
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.