Backyard Minerals

Really Wild Stuff - A Silly Argument About So-Called Resources on Earth and in Space

Delivering asteroidal materials to Earth as manufactured goods does not make much sense. The materials are not beneficiated ores, and manufacturing requires a huge and interconnected network of factories. Gravel or sand is a few dollars per tonne. The asteroids represented by nickel-iron meteorites - mostly metal - are not uniform alloys, and will need purification and blending to make an industrially useful feedstock. Extracting ppm impurities will be expensive and difficult, and will require large amounts of energy and solvents. We will do this someday, feeding the materials to projects in space, but it is unlikely they will ever be competitive with earth sourced materials.

Let's compare "abundant space materials" to something equally silly.

I have a 0.67 acre ( 2711 m², 0.2711 hectare) suburban lot. I (theoretically) own the mineral rights in a narrow inverted pyramid with the apex at the center of the Earth. The earth's surface is 510 million km², or 51 billion hectares - my lot is 5.32e-12 of the surface of the earth, and the same fraction of the total mass of the earth (5.6736e24 kg), so my "share" of that is 3e13 kg. The property is above continental granite, but relatively thin - the Juan de Fuca plate, mostly basalt without beneficiated ore bodies, is subducting down into the mantle perhaps ten kilometers beneath me. There are a few dormant volcanos nearby. The hill that I live on is a long-extinct volcanic cinder cone, topped with glacial till, debris, and silt from the Missoula floods.

Assume this volume is "average" for the Earth's composition, and make the same kind of calculations that are made for materials in asteroids. Assume some magic way to keep the sides of the pyramidal hole from collapsing.

How much energy is needed to cut a hole all the way to a point in the center and lift everything out? How much material is that, and (assuming market prices did not collapse) how much is it worth, purified to commercial grade metals and materials?

See: https://en.wikipedia.org/wiki/Abundance_of_the_chemical_elements and https://en.wikipedia.org/wiki/Prices_of_elements_and_their_compounds and related pages. For prices, check the spot markets.

That's 50 trillion dollars an acre. The entire earth is worth 6e24 or six septillion dollars.
Or nothing, if we wipe ourselves out trying to do this .

• Note: Magnesium is a sizeable fraction of the total. Not because magnesium minerals are particularly valuable, but rather because pure magnesium is difficult to extract from tenacious oxides. Semiconductor grade monocrystalline silicon is vastly more valuable per kilogram than sand, and so are hyperpure crystals of most of the elements listed, but that is even more ridiculous than this calculation.

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Lift Energy compared to Asteroid Re-orbit Energy

The material is in a tall, skinny, inverted pyramid. We will optimistically assume uniform density to make the calculation easier, though this is hugely less optimistic than the absurd idea that we can actually do anything like this.

Assume a perfect sphere with a surface gravity of g, radius R_e , and density \rho. Assume the surface area is A . How much energy d E does it take to lift a slab from depth R with thickness d R to the surface?

The area of a slab at depth R is A_r = A ~ ( R / R_e )^2 . The mass of this slab is d m = \rho ~ A_r ~ dR = \rho ~ A ~ ( R / R_e )^2 ~ d R

Gravity inside a uniform sphere is linearly proportional to radius, so the gravity at radius R_y is g_y = g ~ R_y / R_e . The amount of energy to lift the slab from depth R a distance of d R_y is

d E_R ~ = ~ g_y ~ d m ~ d R = ( g ~ R_y / R_e ) ( \rho ~ A ~ ( R / R_e )^2 ) ~ d_R ~ d R_y ~ = ~ ( \rho ~ g ~ A ~ R^2 / {R_e}^3 ) ~ R_y d_R ~ d R_y

Let's integrate that over R_y from R to the surface R e :

E_R ~ = ~ ( \rho ~ g ~ A ~ R^2 / {R_e}^3 ) ( {R_e}^2 - R^2 / 2 ) = ( \rho ~ g ~ A / 2 ) ( ( R^2 / R_e ) - ( R^4 / {R_e}^3 ) )

The total energy for the entire column of material integrates over R from 0 to R_e :

E_{lift} ~ = ~ \rho ~ g ~ A ( R_e )^2 / 15

The volume of the inverted pyramid is A R_e / 3 , so the assumed-uniform density \rho is the total mass M divided by volume:

\rho ~ = ~ 3 ~ M / ( A ~ R_e )

Producing a very simple result for the whole column:

E_{lift} ~ = ~ 0.2 ~ g ~ M ~ R_e

If g = 9.8 m/s², M = 3e13 kg, and R_e = 6371 km, the total lift energy (at 100% efficiency) E is 3.7e20 J, about 50 minutes of earth solar illumination, or 9 hours of 10% efficient global PV. The energy per kilogram is 12.5 MJ, or about 3.47 kilowatt hours, or a \delta V equivalent of 5000 m/s. Assuming that the heat capacity of earth mantle material is 500 J/kg-K (POMA), a kilogram brought up from the 4000 K lower mantle boundary would add about 2 MJ/kg, a total of 6e19 J, about 8 minutes of total earth surface solar energy.

BTW, the above result is for lifting a tiny fraction of the Earth's mass to the surface. Launching it to escape velocity requires E_{launch} ~ = ~ 1.2 ~ g ~ M ~ R_e . Lift energy is a third of the energy of disassembling the entire earth and spreading it out into space, which is E_{dissassemble} ~ = ~ 0.6 ~ g ~ M_e ~ R_e .

Asteroid Delivery Energy

If the volume of material was shaped into a spherical asteroid, it would have a radius R_A ~ = ~ ( A ~ R_e / 4 \pi )^{1/3} ~ = ~ 1.11 km . But since we need reaction mass, let's assume we are tossing the material off a much larger asteroid with a superhigh acceleration launcher.

Moving a chunk of asteroid from the asteroid belt into low earth orbit is difficult - let's assume we will merely change its orbit from a circular 2 AU to an elliptical orbit between 2 AU and 1 AU, and we will use the atmosphere to slow the material and lower it gently, and that the material somehow remains intact and impacts somewhere safe. We will assume it has the same composition as our hypothetical wedge of earth and is equally valuable.

The 1 AU earth orbits at 30 km/s, and the 2 AU asteroid belt orbits \sqrt{2} slower, 21.2 km/s. An elliptical orbit between 1 AU and 2 AU has an eccentricity e = 1/3, a semimajor axis a = 1.5 AU, a characteristic velocity v_0 = 26 km/s, an aphelion velocity v_a = 17.3 km/s, and a perihelion velocity v_p = 34.6 km/s . So, assuming a relatively small escape velocity from the source asteroid, the asteroid launch velocity is 21.2 - 17.3 = 3.9 km/s, only 61% of the energy of extraction of the wedge from the earth. Note that if both operations are solar powered, the Earth's surface gets about 2.8x more solar illumination.

The asteroidal material arrives at the earth's orbit with a relative velocity of 4.6 km/s, an energy of 10.6 MJ/kg, which is added to the earth escape velocity ( 10.7 km/s ) energy of 57.3 MJ/kg, totalling 68 MJ/kg. That energy would be divided between impact energy and atmospheric heating, but would all become atmospheric heat after the impact energy dissipates. Total energy for 3e13 kg is 2e21 joules, equivalent to 4.5 hours of earth surface solar energy.

Launching mass retrograde from an asteroid will eventually speed it up prograde, increasing the size of the orbit and increasing the delta V necessary for launch (needs analysis). Keeping the asteroid in the same orbit requires balancing mass launched prograde. If the amounts are equal (wasting half the asteroid, preferably the less useful fractions) the total launch energy is doubled.

How many integrated circuits?

MOS FETs will continue to scale and exploit new technologies, but lets assume we stop shrinking at 10 nanometer dimensions, and build finfets out of bars of silicon 10 nm x 10 nm x 30 nm, with metal gates and 4nm thick hafnium gate oxides. We will ignore the conductor metal used, for now, we can probably use abundant aluminum and iron and magnesium if necessary. Plenty of oxygen in air, water, and granite.

The density of silicon is 2.33, HfO₂ is 9.68, and the hafnium fraction of that is 8.40. So the mass of a transistor is:

Transistor manufacture is limited by the abundance of Hafnium, not far more abundant silicon, so my share of the crust can produce "only" 4.3e25 transistors, a mere 40,000 times more than all the transistors in the world in 2015. Priced at 1e10 transistors per dollar, that is $4e15, or 4 quadrillion dollars. Hafnium at current prices constitutes 0.13 ppm of the cost of those transistors. The semiconductor industry's use of this rare metal is too small to impact its price.

How much fission energy?

Assume a technology like the integral fast reactor, that reprocesses fuel and sends the radioactive fission products back through the reactor for neutron bombardment and de-activation. Assuming 200 MeV per fission, 50 MeV lost to deactivation processes, and 33% thermal efficiency, we can expect 5e7 eV per nucleon plant output, or 4.8e12 J/mole.

I don't have numbers for thorium in the mantle, but assume it follows the same 3.33 to 1 ratio to uranium as the crust.

4e7 x 4.8e12 = 1.9e20 J, or 5.3e13 kWh. Half the energy we need for the lift. Of course, if we lift only the crust and mantle, the energy needed will be smaller. OTOH, burying the waste (and everyone elses) at the earth's core might be worth the cost of exposing it. At 4 cents per kilowatt hour, a mere 2 trillion dollars worth of energy.

Conclusion

Overall, the two processes are within an order of magnitude of each other, both delivering gravitationally sorted but otherwise non-beneficiated rock of approximately equal (low) value to the earth's surface. The "core the earth" approach is obviously silly ... besides access to a nickel-rich core, there many disadvantages compared to a mine 20 km deep and 67 acres in area, or a mine 200 meters deep and 6700 acres in area, which would require far less energy to remove. The product is the same: uninteresting rock, unless this was done around a concentrated ore body. Asteroid mining to provide raw materials to Earth is ridiculous.

Asteroid mining to feed raw materials to simple manufacturing processes to produce objects used in the asteroid belt may be less ridiculous - except there is no factory infrastructure there. That infrastructure may grow from nothing to full local capability over hundreds or thousands of years - but please keep in mind that this growth will require new kinds of processes to manufacture new kinds of objects, and that will require a vast accumulation of new knowledge, and a vast infusion of capital to speed it up appreciably (in order to pay for all the mistakes and rapid obsolescence incurred during rapid development). It took 8 trillion dollars to develop the rocket fleet we have - use that to estimate the cost of vastly more ambitious projects.

Terrestrial industrial civilization took thousands of years to develop, with the whole human race participating. Please do not underestimate the effort required to recapitulate the process in a far more challenging extraterrestrial environment. It will never happen without understanding the realities of terrestrial extraction and production; the only advantage we have over our ancestors is accumulated knowledge, if we do not ignore it.