Construction2
Introduction
A construction orbit is a HEEO (Highly Elliptical Earth Orbit) optimized for construction, and synchronous with the launch loop as the earth rotates below. The Earth makes 1 extra turn per solar year relative to fixed space (366.2422 turns per solar year), so the orbit period will be a multiple of a sidereal day (sday), 86164.09 seconds.
note: Actually, the oblate earth ( described by the J_2 harmonic of the gravity field) precesses perigee approximately 3 seconds per orbit, so the actual loopsynchronizing period might be closer to 86161 seconds, with perturbations by the Moon, Sun, Jupiter, etc adding complications to every launch. A professional orbital designer should calculate exact details and optimize launch and throughput with software algorithms. Treat the following as a 0.1% accurate approximation. 
This means that high velocity (~10 km/s) launches from the loop always arrive near the construction orbit apogee, where a tiny delta V ( << 1 m/s ) aligns the orbital planes and corrects for launch errors, and a smallish delta V ( < 120 m/s ) matches launch vehicle velocity with the Construction Station.
Construction orbits with a perigee radius of 8378 km (2000 km equatorial altitude) and apogee radii ranging from 75850 km (for a 1 sidereal day period) to 300205 km (for a 7 sidereal day period) are discussed here. The higher 7 sday orbits require 330 m/s more launch velocity (easy with a launch loop) but less than 40 m/s of "capture" velocity , compared to 114 m/s for the 1 sday orbit.
Cargo vehicles will be entirely passive; they will be equipped with transponders and retroreflectors for precision location, and will have ablative thrust panels for attitude and hyperprecise trajectory control, but the construction station will perform all the trajectory calculations and supply all the thrust. This reduces vehicle aeroshell cost and complexity to the bare minimum.
Each launch loop will support hundreds of construction stations, and there may be dozens to hundreds of launch loops, at different latitudes and longitudes near the equator. The easiest place to construct the first launch loop is at 8 degrees south, 120 degrees west over the eastern Pacific ocean, west of Ecuador and south of San Diego. The weather is boring. The first launch loop will cost billions of dollars ($20B?), mostly for power plant. With a 6 GW power supply, it can launch more than 4 million tonnes to a constellation of perhaps 96 construction stations in one sidereal day orbits.
Much larger launch loops are possible; with space solar power and most of the heat dissipation in the stratosphere, 100 billion tonnes of launch per year might heat the Earth's atmosphere, by 0.01C, while enabling global scale climate measurement and remediation from orbit  a net win for the Earth if used wisely.
Stationsupplied laser thrust
A design driver for launch loop is "no gram left behind". Propellant plumes expelled from rockets are cooled by nozzle expansion, but are still Maxwellian thermal distributions of molecules travelling in many directions at many velocities. From high altitudes, many of those molecules will go into long period orbits around the Earth, with nothing else to run into besides themselves. Light pressure will not push them into new orbits  the scattering cross section for an isolated molecule is small. The best thing to do is to minimize propellant plumes, or else launch them away from Earth at escape velocity.
Cargo vehicles (99%+ of traffic) will be a simple passive shell with passive retroreflectors, and absorptive ablation panels tuned to different laser wavelengths. The construction stations (there will be many) will have precisely aimable high power lasers; many lasers for redundancy, measuring and controlling multiple vehicles in transit.
Loop launch is very accurate, but vehicles may be loaded imprecisely and the atmosphere to exit turbulent, causing small variations in velocity and launch angle. The first interactions will be tiny velocitytrimming ablative thrusts to adjust incoming vehicle trajectories to precisions of centimeter position and micrometerpersecond velocity, from lasers mounted on East station. After that lasers on the construction station (which will be orbiting "nearby") will power more small ablative thrusts.
More velocity tweaks as the vehicle approaches apogee. Prime orbit launches arriving at apogee will be in the same orbital plane and will need no further tweaks. Most vehicles will arrive up to half an hour after prime launch time, and will intercept the construction station farther east, at lower radius. The rotating Earth puts the 8° south launchloop into a slightly different orbital plane. A very small southward (?) thrust ( < 3 cm/s ΔV ) at the midpoint between construction station (and prime orbit) apogee will change planes to put the vehicle on course for a precision arrival a few minutes later.
The propellant atoms from these laser ablation thrusts will probably be emitted isotropically. If the laser Isp is 500 seconds, 5000 m/s, much of the material will escape the earth (escape velocity is 2.3 km/s at 75950 km altitude), but because the ablated material is not directed and expanded by a bell nozzle, the atoms will have a Maxwellian distribution, and some will be slow enough to go into orbit. For a 5 tonne vehicle, a 1 cm/s ΔV, and a 5000 m/s plume, the orbiting remainder of the plume will be a small fraction of 10 grams. However, for 4 million total tonnes of vehicle per year, the total orbiting plume volume is a "small fraction" of 8 tonnes. For 100 billion tonnes of vehicles (someday) per year, that is a fraction 200 thousand tonnes of plume atoms added per year.
If space launch grows as much as I hope someday, both traffic management and plume minimization and mitigation will be very important. Hopefully by that time, lasers will improve and far more of the plume material will escape from the Earth.
Equatorial Crossing
The launch loop is NOT on the equator; the equatorial plane will be crowded with constellations of satellites in circular orbits, and launching through those constellations will be a traffic scheduling nightmare. All nonequatorial orbits are inclined, and cross through the equatorial plane at two positions. Launch loops and the orbits associated with them are a special case; the apogee and perigee velocities are straight eastward (the orbits are not tilted), and the two plane crossings are at the semilatus rectum, whose radius that can be computed from r_{slr} = 2 r_a r_p / ( r_a + r_p ) . Here's a table of crossings:
sidereal day period 
1 
2 
3 
4 
5 
6 
7 
Construction Semilatus Rectum, km 
15091 
15707 
15956 
16095 
16187 
16252 
16301 
re 
2.37 
2.46 
2.50 
2.52 
2.54 
2.55 
2.56 
Loop Launch Semilatus Rectum, km 
11904 
12284 
12435 
12520 
12575 
12614 
12644 
re 
1.87 
1.93 
1.95 
1.96 
1.97 
1.98 
1.98 
More Later
Computing the Plane Change
At the instant of eastbound vehicle release from the launch loop, the vehicle is in a HEEO orbit confined to a plane defined by the velocity vector and the center of the earth. The plane can be characterized by a vector perpendicular to the plane; for a launch loop at 8 degrees south latitude and launching directly to the west, that vector is on an "sin(8°) diameter" circle around the north pole. As the launch point from the loop rotates around the earth, once per sidereal day, the vector rotates around the circle. The "angular distance between two plane vectors is "2 sin(πt/sday) sin(8°)" units. Rotating a velocity vector with magnitude V between planes requires ΔV = 2 V sin(πt/sday) sin(8°).
This is a small velocity, especially near apogee. For a one sday launch orbit ( 907 m/s apogee velocity ) and t = 450 seconds, ΔV = 2 × 907 m/s × sin( 180° × 450 / 86164 ) × sin(8°) = 1814 x 0.03306 x 0.1392 = 4.14 m/s . This corresponds to a launch time (and perigee time) 900 seconds after prime launch, and a much later apogee capture time after prime.
Construction Station Capture
The construction station orbits faster than the vehicle, and overtakes it at around 115 m/s. This is NOT a small velocity tweak; too much to produce cheaply with ablation thrust. How to do that with zero emissions?
Consider the tailhook landing system on an aircraft carrier; the tailhook slows down the aircraft and speeds up the landing cable (and then the windlasses holding the cable, and then the aircraft carrier, slightly. Aircraft carrier decks are relatively short (compared to jet aircraft speeds), so landing aircraft must be slowed FAST.
In space, we don't need a runway, and we aren't concerned with aerodynamics. So, we capture the vehicle with nets. Not small anchored nets, but relatively large nets launched expanded and flat, perpendicular to the relative velocity of the "approaching" vehicle. If we can (somehow) expel the nets flat at 200 m/s and aim them precisely at the front of the incoming vehicle, they will wrap around the vehicle and slow down to the approaching vehicle speed, while it speeds up a bit from the "elastic" collision. The nets might be ringed with peripheral weights so they launch and spread out more precisely, with the weights wrapping around the back of the vehicle. If the vehicle masses 5000 kg, and each net and weight masses 20 kg, then a 300 m/s relative speed "impact" slows the vehicle by 1.2 m/s, with the ΔV spread out over the time the large net takes to wrap around the vehicle. A few hundred nets launched from the construction station in rapid succession will slow the station slightly, but speed up the relatively small vehicle (and its increasingly massive collection of nets a lot more.
A vehicle arriving at 115 m/s and decellerating at 10 m/s² will need 6.6 km to slow down. The net launchers (which electromagnetically accelerate the corner weights) will be arrayed in a ring around the capture point, which is offset from the bulk of the station itself. Nets will be launched from different launchers depending on the needed azimuth angle correction needed.
Vehicle (plus the hundreds of wrapped nets and edge weights) arrives near the construction station slowly. The last few nets are tethered; they bind to the cocoon of nets around the vehicle, and the tethers pull the entire ensemble of vehicles plus tethers back to the station for processing. Tne nets are unwrapped, inspected, and reloaded on the launch ring, or diverted to repair or recycle if damaged. Meanwhile, the vehicle is moved to processing for cargo shell removal and reprocessing, ablation thruster recycling.
Electric thrusters (VASIMR?) will be used to restore station position and velocity. The station thrusters will be multipleredundant and robust enough to survive thousands of capture cycles. They will probably run on argon or nitrogen. Unlike the ablation thrusters, these electric thrusters will be properly expanded and collimated in angle and velocity. Propellant velocities will be fast enough so that almost all the propellant plume exits Earth orbit at faster than escape velocity. There will be lots of solar energy available to power the thrusters for very high Isp; since the station doesn't accelerate (much) these main thrusters can be heavy, robust, and redundant.
The original cargo vehicles that deliver components for the first "embryonic" construction station will use conventional chemical rockets for apogee correction and for subsequent vehicle capture. Stations can be used to "clone" neigboring stations "900 seconds away" with a 27 m/s radial and plane change thrust.
Launch Shell Processing
The cargo shell itself will be made of polyacronitrile fiber tape; the portions of the shell heated during atmposphere exit will be added to the radiation shielding, while the "virgin" fiber tape will be heat processed into superstrong carbon fiber for structural construction uses. The hydrogen and nitrogen produced by this processing will be captured for subsequent uses; the nitrogen for breathing air and also electric thruster propellant.
Details below, needs editing:
The older rendezvous page, Construction1, started from a launch time and an exact associated longitude; calculations were complicated and the radial velocity at capture was large.
A better approach is to start with the capture angle at the construction orbit, which defines an exact radius and radial velocity. From there, find a launch orbit with an apogee, launch angle, and launch time that arrives at the construction orbit with the same radius and arrival velocity. The tangential velocity will be different, on the order of 100 m/s, but we can accommodate that with a "passive net capture system".
So, given the construction orbit, choose an angle from apogee \theta . Construction orbits have low perigees and very high apogees, far beyond GEO, so the apogee velocity is small and the angular velocity very small near apogee. Given a desired arrival time t_{ac} referenced from t_{ac} = 0 at apogee, we can estimate \theta given the apogee radius r_{ac} and apogee velocity v_{ac}
The construction orbit perigee should be well above LEO; assume an altitude of 2000 km above the equatorial radius 6378 km, thus r_{pc} = 8378 km. The construction orbit semimajor axis is defined from the orbit period, which should be a multiple N of an 86164.0905 second sidereal day. From that, we can compute the apogee radius, velocity, and angular velocity \omega_{ac} , then iterate from the desired arrival time to the exact capture angle.

period 
semimajor 
apogee 
apogee V 
ang.freq. 
1 hr angle 
Intercept 

N 
P_c 
a_c 
r_{ac} 
v_{ac} 
\omega_{ac} 
radians 
r_i 
v_i 

sdays 
seconds 
km 
km 
km/s 
radians/sec 
est. 
exact 
km 
km/s 
1 
86164.1 
42164.17 
75950.34 
1.02118 
1.3445E5 
0.048403 
0.048557 
75591.05 
0.19988 
2 
172328.2 
66931.45 
125484.89 
0.63056 
5.0250E6 
0.018090 
0.018104 
125341.34 
0.07978 
3 
258492.3 
87705.01 
167032.01 
0.47745 
2.8584E6 
0.010290 
0.010294 
166948.27 
0.04653 
4 
344656.4 
106247.05 
204116.10 
0.39241 
1.9225E6 
0.006921 
0.006922 
204058.99 
0.03173 
5 
430820.5 
123288.78 
238199.56 
0.33721 
1.4157E6 
0.005096 
0.005097 
238157.13 
0.02357 
6 
516984.5 
139223.02 
270068.04 
0.29802 
1.1035E6 
0.003973 
0.003973 
270034.76 
0.01849 
7 
603148.6 
154291.59 
300205.17 
0.26851 
8.9442E7 
0.003220 
0.003220 
300178.07 
0.01506 
Note that the estimate is pretty good, off by 0.000154 radians for one hour delay and the 1 sidereal day construction orbit; that corresponds to an earth rotation of 2.11 seconds, or about 1 kilometer of distance. Good for capability estimation, too much error for accurate aiming from the loop, which will need precise (millisecond) timing.
Other tables can be constructed for other vehicle capture times.
Starting from the capture time T seconds after construction orbit apogee, we can compute the capture angle L radians to the west of apogee.
Given the launch orbit perigee ( r_{pl} = 6378 km + 80 km loop altitude), the intercept radius r_i, and the intercept radial velocity v_i, we can algebraically compute the launch orbit perigee r_{al} and the angle between intercept and launch perigee L . From those parameters, we can compute the semimajor axis a_l, the eccentricity e_l, the period P_l, then the launch time and launch angle.
The radius and the radial velocity at angle L are given by:
r_i = { \Large { { ( 1  e_l^2 ) a_l } \over { 1 + e_l \cos( L ) } } } ~~~~~~~~~~~~~~~~~~~~~~~~~ v_i = { \Large \sqrt{ \mu \over { ( 1  e_l^2 ) a_l } } } \sin( L )
Let's define two new known parameters in terms of the known parameters r_{pl}, r_i, and v_i$ ,
\alpha \equiv { \Large { { v_i^2 r_pl } \over \mu } } ~~~~~~~~~~~~~ hence ( e_l \sin( L ) )^2 = \alpha ( 1  e_l )
\beta \equiv { \Large { r_pl \over r_i } } ~~~~~~~~~~~~~ hence e_l \cos( L ) = \beta ( 1  e_l )  1
This is enough to create a quadratic equation:
( e_l \sin( L ) )^2 + ( e_l \cos( L ) )^2 ~=~ e_l^2 ~=~ ( \beta  \beta e_l 1 )^2 + \alpha ( 1  e_l )
0 = ( \beta^2  1 ) e_l^2 + ( 2 \beta  2 \beta^2  \alpha ) e_l + ( \beta^2  2 \beta + \alpha + 1 )
Define the coefficients of the quadratic polynomial:
A \equiv ( \beta^2  1 ) ...... B \equiv ( 2 \beta  2 \beta^2  \alpha ) ....... C \equiv ( \beta^2  2 \beta + \alpha + 1 )
And solve for e_l :
e_l = { { B + \Large { \sqrt{ B^2 ~~ 4 A C } } \over { 2 A } } }
The other root of the equation yields e_l = 1 which is nonphysical .
Here's the result of a spreadsheet
mu km3/s3 
398600.44 

re km 
6378.00 

sday s 
86164.09 


prime 
launch 
constr 
solution 

period s 
83238.21 
83229.63 
86164.09 
alpha 
2.0389E4 

omega rad/s 
7.5484E5 
7.5492E5 
7.2921E5 
beta 
8.5156E2 

altitude km 
80.00 
80.00 
2000.00 
A 
0.9927484 

perigee km 
6458.00 
6458.00 
8378.00 
B 
0.15560584 

apogee km 
75950.34 
75944.68 
75950.34 
C 
0.83714253 

semimajor km 
41204.17 
41201.34 
42164.17 
radical 
3.34850072 

v0 km/s 
5.787 
5.787 
5.139 
good e+ 
0.8432575 

eccentricity e from apogee 
0.84327 
0.84326 
0.80130 
check 
0.0E+0 

v perigee 
10.66631 
10.66628 
9.25746 
junk e 
1.00000 

v apogee 
0.90695 
0.90701 
1.02118 
check 
0.0E+0 

intercept theta radians 
0.00000 
0.02299 
0.02724 

intercept radius km 
75950.34 
75836.85 
75836.85 

E eccentric anomaly 
0.00000 
0.07881 
0.08199 

time(theta) sec 
0.000 
879.447 
900.000 

tangent velocity km/s 
0.90695 
0.90830 
1.02271 

radial velocity km/s 
0.00000 
0.11218 
0.11218 

perigee time 
41619.10 
40735.37 
42182.04 
constr4.ods downloadable libreoffice spreadsheet. If you install free libreoffice, you can convert this to any excel format if you wish (there are so many incompatible versions to choose from!) or you can start using libreoffice, save money, and invest it in space instead of Microsoft.