Construction1

This will be merged with ConstructionOrbits real soon now.

captur10.w1 --- added to Construction1

This analysis ignores second-order effects like lunar/solar/Jupiter tides, apsidal precession from Earth oblateness, light pressure, and outgassing, which will add seconds and many meters of perturbations to the precision calculations needed to deliver a vehicle to a rendezvous at an exact position and time. With years of experience and calibration, the launch loop will evolve into a precision instrument for millimeter-accurate delivery of vehicles to destinations.

However, the following calculations will be better than 0.1% accurate, useful for estimating delivery rates and infrastructure requirements. It is better to be roughly right than exactly wrong.

Construction orbit, Act 1

Begin with a construction orbit, with a period of one sidereal day. There are 366.2422 sidereal days per 365.2422 solar days of 86400 seconds, so the construction orbit period is ~~~~ T_c = 86164.0905 seconds

The angular frequency of the construction orbit is ~~~~ \omega_c = 2 \pi / T_c = 7.2921158e-5 radians/second

The Earth's standard gravitational constant is μ = 398600.4418 km³/s². The semimajor axis of a one sidereal day orbit is ~~~ s_c = ( \mu ( T_c / 2\pi )^2 )^{1/3} = 42164.1696 km

The construction orbit perigee should be above the LEO space debris, collision, and litigation belt. Arbitrarily choose a perigee radius 2000 km above Earth equatorial radius: ~~ r_{cp} = 2000 + 6378 km = 8378 km

The construction orbit apogee is twice the semimajor axis on the opposite side of the Earth from the (negative) perigee, so the apogee radius is ~~~ r_{ca} = 2 s_c - r_{cp} = 2 * 42164.1696 - 8378 = 75950.3392 km

Deviating from "standard" treatments, we will choose our reference angle, time, and other orbit parameters in relation to apogee, not the usual perigee. The eccentricity of this orbit is negative: ~~~ e_c = r_{cp} / s_c - 1 = 8378.0 / 42164.1696 - 1.0 = -0.8013005 (no units)

The construction orbit perigee occurs one half a period before time zero: T_{cp} = 0.5 T_c = -43082.0452 seconds

The construction orbit perigee velocity is:

v_{cp} = \sqrt{ ( \mu / s_c ) ( r_{ca} / r_{cp} ) } = \sqrt{ ( \mu / 42164.1696 ) ( 75950.3392 / 8378 ) } = 9257.46 m/s

The construction orbit apogee velocity is:

v_{ca} = \sqrt{ ( \mu / s_c ) ( r_{cp} / r_{ca} ) } = \sqrt{ ( \mu / 42164.1696 ) ( 8378 / 75950.3392 ) } = 1021.18 m/s

There can be thousands of different one day construction orbits with the same apogee, perigee, and period, but arriving at apogee at different times throughout the sidereal day. The constellation will resemble the petals of a flower; while the orbital tracks intersect, two different stations associated with the same launch loop will not pass through the same intersection simultaneously.

Unless the entire construction station is well shielded, the crew must be confined to a shielded core surrounded by lots of hydrogen (food, waste, water, fuel) that can attenuate the dose. If the orbit passes through the radiation belt only once per day, this can be the rigidly scheduled sleeping period, every 23 hours and 56 minutes.

Astronauts are superbly competent people, so they can get their day's work done 4 minutes faster than the rest of us.

If the construction orbit perigee is "sidereal midnight", apogee is "sidereal noon". As we will see, "noon" is when passengers and supplies arrive from Earth, and when passengers and products return for reentry. Launch and entry orbits will have smaller perigees, a smaller semimajor axis, and thus somewhat shorter orbit times than the construction orbit.

Launching from a Construction Orbit to Mars

If an interplanetary vehicle accelerates to a velocity greater than escape velocity at that orbital radius, the excess velocity results in a velocity distant from the Earth of v_{\infty} . ( "Vee infinity" ). For interplanetary launches, higher v_{\infty} reduces travel time, though the arrival velocity can be quite high, which leads to high gee loads entering atmospheres like Mars. The escape velocity at a radius r from the Earth is v^2_{esc} = 2 \mu / r . Given a starting velocity v_r at radius r , v_{\infty}^2 = v_r^2 - 2 \mu / r A small change in v_r can lead to a sizable v_{\infty} . v_{\infty}

Interplanetary launches will begin at the construction orbit apogee, with a small retrograde thrust to drop perigee close to the Earth, maximizing perigee velocity. That maximizes the boost effect of additional delta V at perigee; think of this as adding gravitational energy to the expended propellant, and maximizing the effect of thrust, thus achieving the desired v_{\infty} with a minimum of propellant.

A Hohmann trajectory to Mars results from a v_{\infty} of 2.9 km/s. Consider an "injection orbit" starting at the one day construction orbit apogee radius r_{ca} = r_{ia} = 75950 km. with an injection orbit perigee radius r_{ip} of 6500 km ( 122 km altitude ), for a semimajor axis s_a = 41225 km. The The injection orbit apogee velocity v_{ia} is:

v_{ia} = \sqrt{ ( \mu / s_i ) ( r_{cp} / r_{ca} ) } = \sqrt{ ( \mu / 41225 ) ( 6500 / 75950 ) } = 909.66 m/s . The retrograde injection delta V is 1021.18-909.66 = -112 m/s. Hopefully, the propellant plume will have a velocity on the order of -1800 m/s, so that most of the exhaust plume molecules reenter, that is, have a perigee within the upper atmosphere. 1800 m/s is a low Isp < 180 seconds, so this should use a low energy propellant mixture, perhaps LH/LOX with injected water.

The insertion orbit perigee velocity is:

v_{ip} = \sqrt{ ( \mu / s_c ) ( r_{ca} / r_{cp} ) } = \sqrt{ ( \mu / 41225 ) ( 75950 / 6500 ) } = 10629 m/s

for a v_{\infty} = 2.9 km/s, the interplanetary transfer orbit perigee velocity v_{tp} must be:

v_{tp} = \sqrt{ ( 2.9 km/s )² + 2 μ / 6500 km } = 11448 m/s . . . remember that μ = 398600.4418 km³/s² for the Earth

So, the delta V from insertion orbit perigee to an interplanetary Hohmann to Mars is ( 11448 - 10629 ) = 819 m/s, and the total delta V starting at construction orbit perigee is 931 m/s, far less than the ≈11000 m/s direct rocket launch velocity from the Earth's surface.

A more likely mission launched from a construction is a massive 300,000 tonne "Island One" scale permanent habitat, constructed over years in the construction orbit, launched with a crew of thousands into a permanent Aldrin Mars Cycler orbit. This crew will finish the construction and preparation of a Deimos colonization mission, and deploy hundreds of entry vehicles to slow down in the thin Mars atmosphere, and orbit Mars until they are sequentially landed on Deimos to construct a large Deimos scientific colony. A subsequent visit years later by the Mars cycler will deliver the crew for the colony and their supplies.

MoreLater

The period of the construction orbit need not be one sidereal day, but it should be an integer ratio fraction of a sidereal day.

For example, a 5/3 sidereal day orbit will have a larger semimajor axis than the one day orbit, proportional to the 2/3 power of the period, or 59271.063 km. The perigee might still be 8378 km, yielding a 110164.127 km apogee. Since the launch loop rotates around the Earth once per sidereal day, the apogee of this construction orbit is accessable only once every three orbits, or every five sidereal days. Such orbits are interesting, but not as practical as simple integer ratios. A 5 sidereal day orbit will actually be more useful - one third of the passes through the van Allen belt, and a higher perigee velocity for the start (and end) of interplanetary missions.

A more useful example is a 1/2 sidereal day orbit, with a semimajor axis of 26561.762 km. If perigee is 8378 km as before, apogee is 44745.524 km. This orbit can intersect with loop launches twice per day. However, this accentuates a problem shared by all orbits with "low" perigees; they pass through the van Allen belt during descent and ascent more often and more slowly than higher orbits, so more crew time spent in the shielded core. Indeed, the peak of the outer belt is at 5 Re, about 32000 km, Such orbits are more practical for crew-less automated stations; if the crew cannot work outside shielding, then telepresence is a more practical way to work there.

High multiple orbits: 2, 3, 4 etc. sidereal days, may be useful, but orbits that with perigees near lunar radius may be (WAG) more sensitive to lunar gravitational perturbations. The Moon's semimajor axis is 384399 km, though the perigee and apogee drift due to solar tidal forces - a similar drift will perturb construction orbits and launches to them.

A perigee of 8378 km and an apogee of 384400 km is 196389 km, with an orbital period of 866137 seconds, or 10.05 sidereal days.

So, the most practical construction orbits are likely to be 1, 2, 3 ... 6 sidereal days. For the rest of this discussion, we will focus on launches to the 1 sidereal day construction orbit; this may offer the best compromise between rapid assembly and a higher v_{\infty} .

We will continue the construction orbit description later, after considering how to get there from a launch loop.

Prime orbit, Act 1

From here on out, we will round distance to the nearest kilometer and the time to the nearest second.

The prime orbit is an unmodified launch orbit, with the same apogee time ( 0.0 seconds ) and the same apogee radius as the one sidereal day construction orbit: r_{pa} = 75950 km.

The launch orbit perigee is the altitude of the launch loop (assumed to be 80 km) added to the radius of the Earth: r_{pp} = 6378 + 80 km = 6458 km. The prime orbit semimajor axis is: r_{ps} = 0.5 * ( 6458 + 75950 ) = 42104 km The period of the prime orbit is: T_p = 2 \pi \sqrt{ r_{ps} / \mu } = 85980 seconds.

The trajectory time is half the full orbit, or 42990 seconds, or 11 hours 56 minutes and 30 seconds. Prime orbit perigee time is before zero time, or t_{pp} = -42990 seconds.

That is 92 seconds after the construction orbit passed overhead. The prime orbit perigee velocity is 10551 m/s. 471 m/s is provided by earth rotation velocity, and an additional 30 m/s WAG) is needed for air friction during exit. The estimate loop-related launch velocity is 10110 m/s, or 337 seconds of acceleration at 30 m/s . The acceleration run begins 337-92 = 245 seconds before construction orbit perigee, and 1700 km to the west of the release point.

The construction orbit station is moving at 9.257 km/s, so it will be aproximately 2200 km west of the release point, and 2000 km overhead, or 500 km west of launch loop west station, at an elevation of atan( 500/2000 ) or 14 degrees from zenith. There will be plenty of warning time to abort the launch from the loop, though that can lead to logistic complications for fast assembly.

Prime Launch

The launch loop launch track can be at altitudes between 70 km and 120 km. The drag and heating at 120 km is very low, but that means space debris deorbits more slowly, greatly increasing the debris flux and impact hazard. At 70 km loop altitude, the drag is 4.7x higher and the temperature (in Kelvins) is 11% higher, so the vehicle drag and heating is 5x higher than the slightly higher 80 km, making the vehicle nose shroud heavier and more expensive. We will choose 80 km for this exercise.

The earth rotates one sidereal day in 86141.0905 seconds. At 6458 km radius, that is a rotation velocity of ( 2 π 6458000 / 86141.0905 ) = 471.05 m/s . The launch velocity is the apogee velocity minus the rotation velocity. More precisely, we must add a little bit of extra launch velocity to counteract drag leaving the atmosphere; the amount of drag loss will depend on vehicle aerodynamics and atmospheric conditions, but will probably be less than 30 m/s. Production launches will use realtime atmospheric measurements and exact vehicle measurements to model this effect, resulting in exact trajectories trimmed to millimeters per second.

Prime launch delivers vehicles to the apogee of the construction orbit from the launch site at 6458 km radius (80 km altitude). Before a more versatile capture system is assembled, this is the easiest way to get initial materials to the construction orbit station, without a plane change, once per sidereal day.

The arrival velocity is the prime orbit apogee velocity minus the 1021.28 m/s construction station apogee velocity. That velocity will be faster, and some onboard vehicle thrust will be needed to bring incoming vehicles up to speed. Later, we will discuss cable capture, a way to minimize vehicle cost and complexity by using construction station momentum and makeup thrust.

Orbital characteristics, ignoring apsidal precession, referenced from apogee:

A 142 m/s apogee boost is much smaller than the injection velocity for a circular orbit at 75950 km radius; indeed, it is smaller than the injection velocity into any circular orbit above 275 km altitude. This is why construction orbits are the best destinations from the launch loop.

Reentry after Capture/Rendezvous failure

If the vehicle fails to rendezvous with the station, it cannot quickly resynchronize for a second pass, its orbital period is too fast, and it will arrive after another orbit ( 86141.09 - 85979.72 ) = 161.37 seconds early. It is best to re-enter and re-launch; if a low-cost cargo vehicle lacks a heat shield, re-enter and burn up. A 40 km altitude ( 6418 km radius) is assumed sufficient to lose enough velocity to descend, so a launch orbit abort reentry orbit will look like so:

This can be accomplished with laser ablation. In addition, the vehicle should have a rocket capable of 3 m/s retrograde thrust.

A much larger thrust at can re-enter with the very orbit, starting from the construction station at apogee:

MoreLater === Prime Capture ===

The first table describes a series of increasingly higher altitude construction orbits, with periods that are multiples of sidereal days, synchronizing the orbit with the launch loop as it rotates below.

The first "one sidereal day" orbit will be convenient for the construction of space solar power satellites in synchronous orbits. There may be as many as 96 construction station orbits, spaced around the "sidereal clock", fed by separately synchronous vehicle streams. The quickest return to Earth (say, to return a stabilized accident victim to a hospital on earth) will be from the one day orbit, with a 116 m/s retrograde delta V to drop perigee to 50 km reentry altitude in the atmosphere.

If the launch is not captured by the construction station, the vehicle will be in a shorter period orbit and not synchronize with the station on it's next pass. A 2 m/s retrograde delta V from the launch orbit will also drop perigee to 50 km reentry altitude.

The larger long period orbits will be suitable for the launch of interplanetary missions. They will suffer from larger tidal effects from the moon. Still, if the 5 day orbit at the bottom can be made to work, then the capture delta V will be a mere 40 m/s. After an interplanetary vehicle is assembled in this long period construction orbit and the interplanetary trajectory window opens, the perigee is lowered to perhaps 222 km altitude (6600 km radius) with a 37 m/s thrust at apogee. 2.5 days later at perigee, a 1.1 km/s delta V rocket burn can launch the assembled vehicle with an escape velocity excess vinf = 5.5 km/s for rendezvous with an Aldrin Mars cycler orbit. The referenced paper is for a spartan 75 metric tonne cycler; a vehicle built with 20 4-tonne additions every 5 days over a two year period could mass more than 10,000 tonnes, adding to a vast, shielded wheel suitable for centuries of continuous occupation.

Multiple vehicles per day to a one sidereal day construction orbit

Vehicles can be launched from the launch loop into higher apogee orbits over a ±15 minute window around the prime orbit; they will arrive with a bit more tangential velocity, and far more radial velocity. Vehicles launched before the prime orbit time will arrive with downward radial velocity; vehicles launched after the prime orbit time will arrive after.

I presume the vehicles will be as cheap and as close to passive and uncomplicated as possible, and will arrive near the station to be captured by an active maneuvering tether. They will be perturbed somewhat by the turbulent passage out of the thin remaining atmosphere after a high precision launch by the launch loop, and some way of thrusting them into an exactly precise (to millimeters absolute position and micrometers per second relative velocity) is needed.

• My guess is that a reusable "thrust panel" consisting of a "black" laser-absorbent refractory metal sponge, saturated (somehow) with lots of hydrogen, can be designed to eject that hydrogen at very high velocity when it is smacked by a laser pulse from an orbiting station with very big optics. Wild handwaving, but a laser propulsion genius like Leik Myrabo or Jordin Kare might have some good ideas about how to do it (is Jordin's collaborator Tom Nugent still active?). Depleted panels will be returned to Earth for recharge and resuse, or used for bulk shielding.

  An ablative rubber panel might also work, but might scatter too much material into persistent retrograde orbits and pollute low-Earth orbital space with ram-surface-eroding material. A '''No Gram Left Behind''' policy will be necessary for a permanent gigatonne/year spacefaring civilization.

When vehicles arrive, they will be "lassoed" by a velocity and position-matched loop on a deployed cable. They will pull the cable against a drum and a generator, producing electricity to drive some form of propulsion attached to the station itself. We do not need high ISP, but propellant plumes with tightly constrained velocity profiles will be designed to launch all of the propellant into a retrograde orbit with a perigee below the top of the atmosphere. An inert material like argon might be best, so it does not upset upper atmosphere chemistry too much.

Vehicles arriving at ± 900 seconds will have radial velocities around 660 m/s, or excess kinetic energies of 220 kJ/kg; if that was converted 50% efficiently to propellant kinetic energy, a 10% propellant fraction could be launched retrograde at ≈1500 m/s, an impulse of 150 kg-m/s per vehicle kg, more than enough to restore station momentum "lost" to a vehicle capture. A 670 m/s combined radial and tangential velocity capture is frightening - or perhaps "a mere engineering detail", as antimatter propulsion advocate Bob Forward was fond of saying. I don't know how to do it safely and reliably, but someone reading this may be inspired to learn how.

If the expelled propellant fraction is 10%, we can guess that perhaps one in eight of the incoming vehicles are tankers delivering liquid (argon?) propellant.

MoreLater