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The logistics of sending any kind of vehicle or probe to a specific asteroid in the asteroid belt for mining are so ludicrously expensive in terms of energy/momentum spend that the astrophysics community has largely treated space mining as a joke proposal for science fiction books since about the 60s - and that's before you get to the economics problems (there is virtually nothing in space worth mining and returning to Earth that couldn't be extracted more profitably on Earth in the first place). The case for space mining is a complete, unsalvageable disaster. "Short" version:
I don't doubt that SpaceX will happily take the money of anyone foolish enough to ignore all of the above at their own expense and perform their services as advertised. But their business model does not depend on people with more money than common sense - their big moneymaker is, as others have noted, building a novel telecommunications network with broadband-like performance and selling it to the US government, using novel reusable rocket components that cost orders of magnitude less than the previous state-of-the-art and that can be launched quickly and regularly. I expect their next steps for profitability all revolve around expanding the use of this network to things like surveillance satellites, content providers, etc. I grant that they have some appetite for ridiculous vanity projects like the mars launch stuff, but this is ultimately a manageable marketing expense. But for anyone with some rudimentary literacy in the subject, it should be clear that space mining is not a sustainable business, and as a marketing stunt it is extremely boring (heh).
Why would there be californium on asteroids? You might find some plutonium from interstellar dust (although Earth is again a better source of that), but there's no process that generates californium near enough to Sol that it would actually get here before it decayed.
"On" Earth is plausibly not true, although "around Earth" definitely is. The most concentrated reservoir of antimatter in Sol System is Earth's Van Allen belts. Estimates I've seen are that you can't get the price below a billion a gram making it in particle accelerators due to inherent inefficiencies (currently it's more like trillions), while scoops in the Van Allen belts could conceivably do it for millions.
The elements that are most amenable to asteroidal extraction would be tellurium and the strongly-siderophile metals (Ru,Rh,Pd,Re,Os,Ir,Pt,Au), all of which are strongly-depleted in the crust due to tellurides and native metals (the primary forms of these elements) sinking into the core. Some of these are useful and as such humongously expensive. But, yes, there's the issue that you need to refine them on-site because of the delta-V needed for the return trip, and more generally the Space Bootstrapping Problem where a lot of space industries only make sense if there are other space industries to absorb their products.
A couple of mitigating factors I'll note:
if you were to mine asteroids with people, you would not need radiation shielding for the time on the asteroid, because you could use the asteroid itself - digging deep on asteroids is pretty easy energetically. You still need the radiation-shielded craft to get there, though, which sucks.
mass ratios look far nicer if you bite the bullet and start using nuclear. This sucks for takeoff from Earth because people will get apoplectic, but for things like a return mass driver or an orbital-transfer burn there's less of an issue there. This is getting into issues of "do you really think they're going to let Elon Musk buy a breeder reactor and reprocessing plant", though.
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"Getting places in space is convoluted to begin with, and all trips beyond Earth orbit require carefully calculated momentum assists from various heavenly bodies. The error on these calculations is pretty large relative to the size of most asteroids. It's infeasible to pre-plan a route so specific and so accurate that one could send a spacecraft to a specific asteroid in the asteroid belt once, let alone reliably at different times. The energy and time cost for any such trip would be enormous as well."
I'm already lost on your first point. NASA has already done this. For both asteroid belt asteroids and near earth asteroids. If you don't swap your inches and centimeters then yes you can go to a specific asteroid. It is not too complex, we understand orbital mechanics.
There is no reason to mine in space while everything we need is much cheaper to obtain on earth. It’s unnecessary science fiction to build “industrial tech” settings that include asteroid mines and lunar helium farming or whatever.
This seems to presume that this state will persist into some unspecified eternity. Politics alone can make space mining plenty competitive. It's the same reason why "peak oil" predictions kept getting bodied.
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There is no need to mine the new world when everything is much cheaper to obtain in Europe. Some have argued that most colonies actually did cost their home countries much more than they brought in, but one can't argue with the results. THE USA!
The new world actually did offer resources not available, or available only in very short supply, in Europe.
Same could be said for space. You can't find 6 trillion dollars worth of platinum laying around here anywhere!
And if there’s ever a demand for $6 trillion worth of platinum the same way as for sugar and tobacco, you’ll have a point.
You don't know what you can use it for till you get that much! Heck tobacco wasn't even a thing before they brought it back! Certainly don't need sugar or tobacco, people just like them, pure luxury goods. Platinum has actual uses. From pure utils, tobacco and sugar have been huge negatives for human health and society including the horrible conditions on sugar plantations.
Europe didn’t need tobacco, sugar, chocolate, indigo, and all the rest. But upon discovering these things, there was enough upfront demand to convince the merchant class that new world colonies would earn a profit.
What demand for an entire asteroid’s worth of platinum is up-front enough to convince someone to fund anything as expensive as asteroid mining? I guess you could postulate some new room temperature superconductor/cold fusion process relying on it, but then you’re firmly in the realm of science fiction.
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Just from the sheer energy inputs, space mining rockets will not compete with terrestrial dump trucks while there are any appreciable mineral reserves on earth. When industrial civilization reaches out for asteroids, it will be "resorting" to spice mining, not "advancing" to space mining.
There is also the matter of $5 trillion platinum asteroids and the like, but the price of such metals would crater if you tried to sell any appreciable amount.
Space mining may become advantageous if we have significant material demands in orbit.
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THE SPICE MUST FLOW! It all depends on if how we value energy in the future. We are sitting next to a basically infinite supply of it.
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