NoGramLeftBehind

No Gram Left Behind

An additional space launch requirement important to me is "no gram left behind".

A rocket exhaust plume (neglecting small trim thrusts) should not end up in a semi-permanent Earth orbit. The plume should either fall to Earth, or escape into interplanetary space. Long duration retrograde plumes are especially nasty. What stops them is another spacecraft, and if that is an SSPS or space laser mirror, bye-bye reflectivity.

You can estimate plume propellant velocities by subtracting average exhaust velocity from vehicle velocity over the span of the delta V burn, then adding the Maxwellian thermal velocity distribution at the exhaust temperature.

Which sounds more complicated than it is, mostly you are looking for "soft boundaries" in the 3D velocity distribution for escape and for re-entry, then estimating the fraction of the plume within those boundaries.

Raleigh scattering teaches us (sadly) that the amount of momentum accumulated by a molecule from sunlight will be small; in day-period Earth orbits, a tiny momentum gain in the half-orbit away from the Sun will tend to balance the momentum loss orbiting towards the Sun. The small differences will add statistically, and the molecules will eventually "random walk" their way to reentry or escape. That might take millenia.

If the atoms ionize, they can add to the van Allen belts and rev up to MeV energies.

What's already there?

One day ConstructionOrbits will range from 8378 to 75950 km radius, or 1.3 to 12 Re . According to this Southwest Research Institute webpage about the geocorona, neutral hydrogen density ranges from 1000 atoms per cubic centimeter at the inner edge of the ring current ( 4 Re? ) to less than 100 at geosynchronous orbit, 6.6 Re. That is 1e9 H/m³ or 1e18 H/km³ at 4 Re and 1e17 H/km² at 6.6 Re. Per Re unit, that is 2.6e29 H/Re³ at 4 Re and 2.6e28 H/Re³ at 6.6 Re and above. In more manageable units, that is 433 kg/Re³ amd 43 kg/Re³ respectively.

As a wild guess of an upper limit, assume the density drops linearly (it is probably an inverse exponential) from 433 kg/Re³ at 4 Re to 433 kg/Re³ at 6.6 Re, then remains constant from 6.6 Re out to the 12 Re. In the inner linearly decreasing shell, the density formula is

\rho(R) ~ = ~ 1033 ~ - ~ 150 R ~ kg/Re³

The mass of a thin shell is: \partial M (R) ~ = ~ ( 1033 - 150 R ) ( 4 \pi R^2 ) \partial R

So M(R) ~ = ~ {\Large \pi \over 3 } ( 4133 - 450 R ) and M(6.6) - M(4) ≈ 200 tonnes

The outer shell volume between 6.6 Re and 12 Re is 1440 Re³, so the mass of that shell is ≈ 600 tonnes. The total mass including helium, and a more realistic density dropoff, is very likely less than 1000 tonnes. "Standard" sea level air is 1.225 kg/m³, so this is less mass than a cube of air 100 meters on a side, and vastly less than the entire 5e15 tonne Earth atmosphere.

The total mass in orbit is on the order of 10,000 tonnes. About 500 are geostationary satellites. If they were all like the largest 7 tonne Telstars, that would be 3500 tonnes in GEO. A simple geostationary transfer orbit (GTO) might have an apogee of 42164 km and a perigee of 7000 km, with an apogee velocity of 1.64 km/s. Geostationary orbital velocity 3.07 km/s, so a simple apogee kick is a delta V of 1.43 km/s . Provided by a rocket with a 3 km/s exhaust velocity (like RP-1/LOX), the propellant plume might be 60% of the mass of the satellite, or 2100 tonnes of propellant, ejected retrograde between velocities of 1.36 km/s and -0.1 km/s.

From that altitude, almost all of the plume will reenter. However, the plume will be hot, and incompletely expanded, so some of the molecules emitted early in the burn may have thermal velocities higher than 100 m/s, and will have perigees higher than reentry altitude. They will stay in orbit until they run into something; hopefully a gas molecule in the thermosphere, but possibly a satellite.

Papers

R. Richard Hodges Jr, 1994 JGR Vol 99, "Monte Carlo simulation of the terrestrial hydrogen exosphere

• page 23,233 cites Tinsley 1975, 1.3e27 hydrogen/s (2.2 kg/s), or 4.1e34 hydrogen/y (68,000 tonnes/y )

• page 23.237, ballistic lifetimes: F10.7 80 > 2.3 days, F10.7 230 > 1.0 days

• page 23.237, satellite lifetimes: F10.7 80 > 1.6 days, F10.7 230 > 1.0 days

H. Fahr and B Shizgal 1983 Reviews Geophysics and Space Physics Vol 21 "Modern Exospheric Theories and Their Observational Relevance

• page 77 fig 2 vertical temperature distribution
• page 80 fig 4a velocity escape/orbit/reenter diagram, what I call the "rotten apple"
• page 81 table 2 types of exospheric particles

N. Ostgaard, S. B. Mende, et. al. 2003 JGR vol 108 "Neutral hydrogen density profiles derived from geocoronal imaging

• page SMP 18-9 Eq 13 n(r,t,\phi) = C(t) \left[ n_1(\phi)e^{-r/\alpha_2(\phi) } ~+~ n_2(\phi)e^{-r/\alpha_2(\phi) } \right]

• C(t) is determined empirically and varies between 0.8 and 1.3 over the year [ khl note: over the timespan considered in this paper ]

• \phi is the solar zenith angle

Table 2: Hydrogen Density Parameters for Equation 13

• page SMP 18-10 "Contrary to some earlier reports [ e.g. Wallace 1970. Betraux 1973] we do not find an evidence for depletion due to charge exchange with solar wind protons at high altitudes.

• page SMP 18-10 Conclusion (5) Our results are only valid above 3.5 RE

All in all, errant molecules in Earth orbit are corrosive spacecraft killers. Multi-decade accumulations shed from a gigatonne-to-orbit-per-year spacefaring civilization will imprison that civilization long before we complete the process of colonizing the solar system, much less spreading into the galaxy.