Asteroid flux or: How to make a planet perfect for mining

Question

One of the cool things about the Moon is that the far side has a thicker crust that the near side.¹ One theory explaining this is that the Moon was hit by an object, possibly a moonlet created by the remains of Theia as per the Giant Impact Hypothesis (while the Moon was still forming). The object hit in such a way that more material was deposited than was blown away, leading to the thicker crust.

Let's say I own an interstellar mining company, and I've come across a system with plenty to spare. There are four rocky planets orbiting a central Sun-like star. There's also an asteroid belt at between 3 and 4 AU, with a total density (i.e. number density of asteroids multiplied by the mass of each asteroid) of about three times that of the asteroid belt in our Solar System.

I don't want to mine each asteroid directly because that would require some major infrastructure problems. However, I've figured out that I might be able to divert some of the asteroids towards the planet, which lies at 2.5 AU. If I get it just right, I might be able to move some asteroids close enough to leave a lot of mineable material. Then I can swoop in and set up shop.

The issue is, too many impacts could cause a whole bunch of issues for the planet, making it worthless. And too many impacts could damage the material already there. At the same time, I want to get as much material as possible deposited onto the planet.

Assuming I can divert a large number of asteroids away via some as-yet-undetermined (but not precise) method, how many should I send in the direction of the planet to maximize my yield? Note that I'll need a large asteroid flux (in terms of number density) because many could and will completely miss.

A potentially useful paper is Asteroid and comet flux in the vicinity of Earth (Shoemaker et al, 1990).

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I'd like answers to focus on how much material could be deposited on each impact, orbital perturbations of the planet, regional distribution of impacts (due to planetary rotation) and how they can impact mineability of deposited material, and asteroid flux. Answers discussing only one or two factors, however, will be much appreciated; you don't need to cover everything to write a high-quality answer.

Answer 1

TL;DR: It is unlikely that this is the best option

Let's look at some numbers that describe the amounts of energy needed for some of the things you're talking about. First, let's assume that both the planet and the asteroids are in a circular orbit around the star. Also, let's say the star and the planet have the same mass as our sun and the Earth, respectively. Now, some formulas:

$$M_p = 5.972*10^{24} kg$$ $$M_s = 1.989*10^{30} kg$$ $$G = 6.67*10^{-11}\frac{m^3}{kg\cdot s²}$$ $$E_{orbit} = -\frac{GmM}{2r}$$ $$1 AU = 1.495*10^{11}m$$

These are $M_p$ the mass of the planet, $M_s$ the mass of the sun, $G$ the gravitational constant, $E_{orbit}$ the potential energy of a mass $m$ orbiting around a mass $M$ at radius $r$, and $1 AU$ in meters. Using these, we can determine how much energy it would take per kilogram to drop an asteroid onto the planet and then cost per kilogram to remove it from the system.

The potential energy of an object orbiting the sun at 2.5 AU is $-\frac{GmM_s}{2*2.5AU}=-1.775*10^8J/kg$, while at 3 AU it is $-\frac{GmM_s}{2*3AU}=-1.479*10^8J/kg$, for a delta of $2.958*10^7J/kg$. That means to move a $10^9kg$ (pretty small) asteroid you'd need $2.958*10^7*10^9=2.958*10^{16}J=8.217*10^9kWh$.

A nuclear reactor generating 1,000 megawatts (i.e full scale terrestrial nuclear reactor) would need close to a year to generate that much energy if there is no waste energy. The Tsar Bomba generated about that much energy total but most of the energy would just be lost to space, it would be very hard to aim, and it's likely that the energy would shatter the asteroid anyway, leaving very little to fall on the planet. Once you'd gotten the asteroid (or most of it) to the planet, you'd need that much energy again to move that much mass back from 2.5AU to 3AU, plus about that much again to account for escaping the gravity well of the planet.

You'd need to use a ridiculous amount of energy moving asteroids (even ones nearer the planet) down to the planet and then getting the mass back out, especially with how much loss you'd get from the asteroids missing the planet or not being entirely salvageable. It would be far easier to build some mining facility on an asteroid and then pack up and move the entire facility to the next asteroid once you're done. Even if you're using applied phlebotinum to move the asteroids, you'd have an easier time applying the phlebotinum to moving the facility.

*Note: My calculations for dropping the asteroid onto the planet assume you'd try for a relatively gentle collision with the planet. You can save some initial energy by putting it into an elliptical orbit, but then the impact will be more severe and your aim has to be much better. If those aren't an issue, then you can save 50% energy on the initial drop to the planet. Also, I don't expect that these completely correct, but it at least gives you an idea of the scale required.