# Early and Late Launch to the One Day Construction Orbit

Burst launches enable high vehicle delivery rates from the launch loop to ConstructionOrbits . Vehicles launched to the prime orbit arrive directly at construction orbit apogee. Vehicles launched soon after that perform a small ( < 10 m/s ) Plane Change before synchronized Rendezvous and Capture. This article explains the process, and why vehicles launched before the prime orbit launch time are not easily captured.

A fully powered, minimum launch loop can launch as many as 80 five tonne vehicles per hour. The most direct launch to a station in a construction orbit is synchronized to an arrival when the station reaches apogee. This is the "prime" launch time, occuring once per (86164.09 second) sidereal day. The destination cargo orbit is "synchronized" to the "celestial sphere", not the position of the Sun, so the Earth rotates a little extra each day relative to the sky, in relation to the "moving" Sun.

- A small additional complication is the oblateness of the Earth; the equatorial bulge modifies an elliptical construction orbit slightly, shifting apogee and perigee eastward by about 0.012 degrees ( 3 seconds of "extra" rotation ) per orbit. This turns the axis of the orbit one full turn around the sky approximately once per 80 years. Mission planners must consider this and other effects when designing the software to plan ultra-precise launch loop trajectories, along with tidal effects from the Moon, Sun, and Jupiter, as well as light pressure effects on vehicle surfaces.
We will not worry about these small second order effects here, beyond noting that they exist, and must be compensated for with high precision measurement, launch and thrust control. The LAGEOS laser geodynamics satellites demonstrate that we can measure Earth orbits to micrometer precision over mega-meter distances; if we can avoid large turbulent rocket thrust, we can design space trajectories with 9 digit precision.

With some additional complication, we can launch burst groups of vehicles to rendezvous the same destination station in timeslots after the prime launch time (and possibly before), greatly increasing delivery rates and facilitating rapid construction of very large objects. If the construction station can capture 20 five tonne vehicles per day over a 15 minute timespan, and the vehicle payload fraction is large, that is a net growth rate exceeding 300,000 tonnes per year. After two years, an assembled station with a structural mass of 500,000 tonnes and 100,000 tonnes of propellant could resemble Gerard O'Neill's Island One, a complete community with 10,000 inhabitants. This could be launched into an Aldrin Cycler orbit to Mars/Deimos, and be the main research and construction facility for Deimos settlements and robotic Mars surface exploration missions, carrying research equipment as large as the Advanced Light Source at Brookhaven, for subnanometer probing of biomolecule candidates discovered on Mars.

A fully powered minimum sized launch loop could support 100 of these constructions, spreading cycler and destination habitats throughout the asteroid belts and eventually the Oort Cloud. Larger loops could supply the construction of vast habitats much more quickly. All this is enabled by high volume daily burst launches, launching after prime launch (and perhaps before).

### Plane Change

Launch from the first minimum-sized loops is affected by extreme weather. The weather is benign (and boring to meteorologists) at 8°S latitude, 120°W longitude, due south of San Diego, so that is a good place to deploy the first large launch loops.

A loop at 8°S latitude will launch into a tilted orbital plane, with a perpendicular at 82°N latitude. The perpendicular rotates around the sky once per sidereal day. A launch before or after "prime time" will be into a slightly different orbital plane than the prime orbit and the construction station, necessitating a small plane-changing thrust into the construction station orbital plane before arrival. Plane change thrust is small but necessary, without it, a 900 second early vehicle may arrive 4 kilometers south of the construction station.

How much thrust?

Imagine a "velocity globe", with a radius equal to V_0 , the velocity of the vehicle near launch orbit apogee, perhaps 900 meters per second. The prime launch velocity vector can be represented as a spot on the velocity globe at 82°N, "prime construction station meridian". Other launches throughout the day can be represented as spots around the circle.

The "angular radius" of the circle is \sin{ \phi } , and the "velocity radius" is V_0 \sin( \phi ) , where \phi is the latitude of the launch site (and the angular distance of the circle from the "velocity pole") in **radians**. 8° latitude is 0.1396 radians latitude.

The angular spacing around the angular circle \lambda is the time difference divided by the sidereal day s (86164.09 seconds), multiplied by 2 \pi for radians or 360° for degrees. The velocity change is the chord distance across the circle, proportional to 2 \sin( \lambda / 2 ) .

The plane crossing occurs halfway between the apogees (or the perigees) of the launch and the construction orbits. So, the velocity plane change, a north/south thrust applied at the plane crossing, is (using radians for your computer):

\Delta V ~=~ 2 ~ V_0 ~ \sin( \phi_{radians} ) ~ \sin( ( \pi \Delta t / s ) ~~~~ . . . using radians, or

\Delta V ~=~ 2 ~ V_0 ~ \sin( \phi_{degrees} ) ~ \sin( ( 180° \Delta t / s ) ~~~~ . . . using degrees for your calculator and medieval brain.

The angles are small, so we can approximate \sin( \phi_{radians} ) \approx \phi_{radians} and the equations as L

\Delta V ~\approx~ 2 \pi V_0 \phi_{radians} ( \Delta t / s ) ... using radians

\Delta V ~\approx~ ( \pi^2 V_0 \phi_{degrees} / 180° ) ( \Delta t / s ) ... using degrees

For \phi_{degrees} = 8° latitude, V_0 = 900 m/s, and s = 86164.09 seconds, the equation simplifies to ** ΔV ≈ 9.134e-4 m/s² × Δt **. For Δt = 900 seconds (15 minutes), ΔV ≈ 8.22 meters per second.

The propellant plume for this thrust will go into an orbit slightly inclined from the launch orbit; vectoring it retrograde will boost the vehicle velocity a bit, while insuring the plume re-enters into the atmosphere. This conforms to the No Gram Left Behind Principle required for clean orbits in a gigatonne launch economy, but this is another analytical complication which we will not go into here.

The plane change should be made near apogee, where the velocity (and thus the radius of the velocity globe) is much smaller than the radius of a velocity globe near launch perigee. A vehicle with wings, launched from a lower altitude, could make limited plane changes near perigee using aerodynamic lift. However, wings are expensive, and so is the drag of a lower altitude launch. Perhaps possible, but not recommended.

### Capture vs Rendezvous

Today, vehicles arriving at ISS perform a slow rendezvous, matching both position and velocity with ISS, "hovering" and performing systems checks, then approaching at centimeters per second. This is safe and simple and "the way we've always done it", but it requires a complex and expensive vehicle, and takes many hours. ISS is fragile, and a vehicle collision could destroy it, so the capture technique described below would be completely unacceptable.

The first arrivals during the initial stages of construction station assembly must be similarly cautious; expensive, reusable vehicles with complex rockets and maneuvering systems delivering smaller payloads. However, after hundreds of payloads are assembled, the resulting station will grow to have a thick radiation-reducing shell, and can be robust to payload collisions.

Arriving vehicles can be captured by separate, armored, uninhabited, co-orbiting "small" vehicles controlled from the construction station: **capture tugs**. The capture tugs will be equipped with reusable, robust and repairable small-delta-V rocket propulsion systems; they will be frequently refueled and ΔV per capture cycle will be relatively small.

Capture tugs can launch capture "lassos" at the "tailhooks" of incoming vehicles, slowing them down (over kilometers) and then reeling them in; an "aircraft carrier cable arrest system" without the need for a deck or a hull. There may be a simpler way; the goal is to make the vehicles as simple as possible, and the massive and expensive construction station as safe as possible from malfunctioning or misaimed vehicles. The capture tugs encounter the risk. They will only "talk to" the construction station, and relatively immune to malicious software exploits (I worry about this for the International Space Station; an incoming vehicle's hardware may be 99.99% safe, but rogue software can turn it into a kinetic kill weapon against the fragile ISS).

The capture tugs use the arriving vehicle's kinetic energy difference, plus stored energy (batteries? They can be **heavy**), to add energy to propellant and fire rockets (forward thrust, retrograde plumes) and rendezvous with the main construction station. In conformance with the No Gram Left Behind principle, the rocket motors will be designed (somehow, much handwaving) to constrain the maximum retrograde velocity of the plume (not just the average, but the statistical outliers) so that the entire plume reenters when the molecular orbits descend near the Earth.

The relative rendezvous velocities are calculated next, then we will return to operational details of the capture tug system.

### Rendezvous

The launch vehicle and the tug capturing it must cross trajectories simultaneously for a capture to be possible. If they are coplanar, their ellipses will cross if the apogee of the launch orbit has a larger radius than the construction orbit at the same angle as the that apogee.

- Note: a slightly lower apogee may also work, with the launch orbit barely "kissing" the construction orbit at the same angle; finding this slightly lower apogee is a difficult (and unnecessary) process.

### Capture