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, 86164.09 seconds.

• note: In reality, the J_2 term describing the oblate earth precesses perigee perhaps 3 seconds per orbit, so the actual period might be closer to 86161 seconds, with perturbations by the Moon, Sun, Jupiter, etc.. It's complicated; let a professional orbital mechanic calculate the exact details, and treat the below 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 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.

The vehicles can be entirely passive; they will be equipped with transponders and retroreflectors for precision location, and will have ablative thrust panels for attitude and hyper-precise 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 remediation from orbit.

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Station-supplied thrust

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Details below need 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.

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

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.