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. Calculate the semimajor axis of a one sidereal day orbit:

s_c = ( \mu ( T_c / 2\pi )^2 )^{1/3} = 42164.1696 km

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 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 perigee, not the usual apogee.

The eccentricity of this orbit is negative:

e_c = 8378.0 / 42164.1696 - 1.0 = -0.8013005 (no units)

The perigee of the construction orbit 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 / 42164.1696 ) ( 75950.3392 / 8378 ) } = 9257.46 m/s

And the construction orbit apogee velocity is:

v_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 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.

If a 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}

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