Confused: what relationship does the launch cost from earth have, to the weight of returned material from asteroids/mars?

Rocket equation.

There's presently no way of constructing or provisioning a rocket entirely in space without first lifting materials (or the entire craft) from Earth. Should there be, whatever costs are associated with that will shift the equation here.

In the meantime, you're stuck with the reality that whatever mass you plan on transporting to the asteroid, and whatever mass is required to haul back your loot, needs to be boosted from Earth's surface to LEO.

The tyranny of the rocket equation dictates what delta-v costs you. Trips with burns at both ends are vastly more expensive than those with burns at only one. So, yes, aerobraking (on return to Earth) is a highly cost-effective method, but that's going to require budgeting for the mass of your reentry shielding and landing mechanism (likely parachutes).

Ultimately, you're looking at a high-speed, high-temperature reentry, atmospheric slowing, and an ultimate soft-ish landing of whatever you've recovered.

While ion rockets have been proposed as vastly more mass-efficient than chemical rockets, existing designs based on xeon rely on an element whose prevalence in known space environments is quite low. The one exception is Jupiter, but that's the second deepest gravity well in the Solar System:

Within the Solar System, the nucleon fraction of xenon is 1.56 × 10−8, for an abundance of approximately one part in 630 thousand of the total mass.[53] Xenon is relatively rare in the Sun's atmosphere, on Earth, and in asteroids and comets. The planet Jupiter has an unusually high abundance of xenon in its atmosphere; about 2.6 times as much as the Sun.

http://en.wikipedia.org/wiki/Xenon#Occurrence_and_production

Sure I get that. Landing via parachute remove almost all the return cost. Leaving the cost of a mechanism in space that can run for years, delivering asteroid metal to earth. Its disingenuous to claim a fixed startup cost prevents profiting from what essentially becomes an industrial infrastructure. It costs billions to create a new oil refinery, yet we do it all the time.

You still have:

Earth -> LEO

LEO -> Asteroid transfer orbit (ATO)

ATO -> asteroid capture orbit (ACO)

ACO -> landing

Takeoff -> Earth transfer orbit (ETO)

ETO -> Earth Capture Orbit (ECO)

ECO Earth de-orbit burn(s)

Your delta-v for return may be low (60 m/s), though most are higher (see below), but that relies on finding near-earth asteroids with favorable mineral characteristics. You can reduce the mass you're returning _if_ you can refine or reduce it on-site, but that requires additional mass to be transferred out.

Orbital mechanics are far from my forte, but none of this comes cheap, and you're still stuck with costs in the order of $5,000 - $10,000 / kg for Earth to LEO. Which includes the mass of your vehicle, its fuel, and any mining equipment your lugging around.

https://en.wikipedia.org/wiki/Delta-v_budget#Near_earth_obje...

A catalog of 11,834 NEOs as of yesterday maintained by NASA / JPL shows a minimum delta-v of 3.8 km/s and a high of 26 km/s. Mean is 7.97 km/s, median is 7.1 km/s.

http://echo.jpl.nasa.gov/~lance/delta_v/delta_v.rendezvous.h...

This isn't a matter of fixed startup costs. Every fucking mission incurs the transfer fuel costs.

Every mission doesn't have to start from the ground. We don't build a new refinery for every tanker. Make your factory in orbit (or better yet- near the asteroid field). Send the refined metal back.

The deal is, deflect just ONE asteroid to earth orbit, and that's maybe more metal than our civilization has mined so far. The potential is, a new order of society here on earth. The cost - some billions.

"Every mission doesn't have to start from the ground."

I've already addressed that. We do.

"Near the asteroid belt" is meaningless. "The asteroid belt" is huge, and "near" in this case would mean "within a low delta-v orbit". Either way, you're shuttling raw ore or the refining equipment.

There's actually an avenue you haven't proposed: utilizing the asteroid directly for propulsion. There are a few possibilities, including laser ablation (there's a recent PhD thesis on this proposal: http://theses.gla.ac.uk/5219/1/2014GibbingsPhD.pdf ), "pebble drives" in which a mass launcher ejects loose material from the asteroid directly (creating potential collision hazards for other craft, though space is big, really mind-bogglingly big), or solar sails using sunlight to alter orbits gradually over a long period of time.

I'm pretty skeptical on all counts.