In several of the methods for designing planetary orbits explored in previous blogs, we did see orbital gaps of 2.0 AU or greater which would be ideal locations for an asteroid belt.
Extrapolating From Our Local Example
Since very little is known about asteroid belts beyond the Solar System, we can invoke The Omega Argument and extrapolate from the asteroid belt in our own back yard.
- Our local asteroid belt lies in the gap between the orbits of Mars and Jupiter; this gap is ~3.68 AU wide.
- The belt extends over a range of ~[2.3, 3.3] AU, having an average width of 1.00 AU.
- The width of the belt is 1.00 ÷ 3.68 = 0.27 times (or 27%) the width of the orbital gap between the planets.
- The midrange of the belt falls at (2.3 + 3.3) ÷ 2 = 2.8 AU.
- This midrange orbit is closer to Mars than to Jupiter (1.28 AU from Mars to the middle of the belt, compared to 2.40 AU from the middle of the belt to Jupiter); these distances equate to ~35% and ~65% of the total gap width.
- The gap between Mars and the inner edge of the belt is 0.78 AU; between the outer edge of the belt and Jupiter, the gap is 1.9 AU; so, the inner gap is 41% the width of the outer gap.
We can establish some generalities from our assessment of the Solar System asteroid belt:
- The belt width is about one-third of the width of the gap between the two planets.
- The midrange orbit of the belt lies closer to the planet with the smaller mass. This makes sense, because the larger mass planet will have either captured, pulverized, or ejected from the system any small bodies that strayed too close. The lesser mass planet will do so, as well, but not as often, and the asteroids would have to come much closer to it.
W = Width of the belt in AU;
M = midrange orbit of the belt in AU;
OO = the orbit of the outer bracketing planet in AU;
OI = the orbit of the inner bracketing planet in AU;
MO = the mass of the outer bracketing planet in Earth units;
MI = the mass of the inner bracketing planet in Earth units;
Ei = inner edge of the belt in AU;
Eo = outer edge of the belt in AU.
Note that using natural logarithms instead of common logarithms always produces values 2.30259 times, or LN(10), times larger.
These equations don't produce the precise parameters for the Solar System asteroid belt, but they generate a close approximation:
The Belt Width is:
This also reveals that Planet 8 cannot be the first gas giant, because it orbits closer than the Frost Line of this system (and it's too far out to be a hot Jupiter). The title of First Gas Giant, then, has to be transferred to Planet 9. With the values for the asteroid belt added, our orbital table now looks like this:
Asteroids in the Habitable Zone?
Note that there is no rule that says that star systems cannot have more than one asteroid belt, so the gaps between planets 10 and 11 and Planets 11 and 12 could also contain asteroid belts, the sizes of which would be calculated as above.
A Massless Alternative
- It necessitates deciding on the masses of at least two planets in the system at a point in the process where this may not be desired (or feasible); and,
- If those masses should change at some later time, the parameters of the asteroid belt will have to be recalculated.
1. Find the Gap Width
a. Calculate one-third of the distance between the bracketing planets.
2. Find the Midrange Orbit
a. If the inner planet is the less massive, calculate the midrange orbit of the belt by adding the value calculated in Step 1a to the orbit of the inner bracketing planet.
b. If the outer planet is the less massive, calculate the midrange orbit of the belt by subtracting the value calculated in Step 1a from the orbit of the outer bracketing planet.
3. Find the Belt Width
a. Calculate a value between 25% - 33% of the gap between the two planets, or
b. Use the belt width calculation from above, since it does not depend on knowing the masses of the planets.
4. Find the Belt Inner Edge
a. Subtract half of the value calculated in Step 3a (or Step 3b) from the midrange orbit calculated in Step 2a (or Step 2b).
5. Find the Belt Outer Edge
a. Add half of the value calculated in Step 3a (or Step 3b) to the midrange orbit calculated in Step 2a (or Step 2b).
Here are the equations for this process:
G = one-third of the width of the gap between the two bracketing planet orbits, in AU;
W = the width of the asteroid belt in AU;
M = the midrange orbit of the belt in AU;
Ei = the inner edge of the belt in AU;
Eo = the outer edge of the belt in AU
Note that one can use the Belt Width equation from above in this process, since it does not rely on the masses of the planets. In general that equation will produce narrower belt widths than the alternative equation in this process.
Note that the asteroid belt is wider than the previous one (by 0.754 - 0.539 = 0.215 AU), but still fits between the bracketing planets.
Density of asteroid belts
In the Solar System's asteroid belt, fully half of the mass (about 4% of the mass of the Moon, in total) is contained in just four bodies (Ceres, Vesta, Pallas, and Hygiea), and fully a third is contained in Ceres, alone. The rest of the mass is divided between about 200 bodies of about 100km in size, 1-to-2 million bodies around 1km or larger in diameter, and many millions of smaller bodies, down to the size of dust particles, created by the rare collisions between asteroids.
There are literally thousands and tens-of-thousands of kilometers between bodies in the asteroid belt. Though collisions do happen, they are comparatively rare. "Collisions between main-belt bodies with a mean radius of 10 km are expected to occur about once every 10 million years." 
Between 1972 and now, no fewer than 12 probes have passed through the asteroid belt without incident, including the Pioneer and Voyager pairs, Ulysses, Galileo, Cassini, and New Horizons. Several of these took images of asteroids as they passed by, but none were ever in any danger of accidentally impacting one.
As recent exploration (the Dawn, NEAR and Hayabusa missions) have shown, a great deal of navigating and course-correcting is necessary to arrive at any given asteroid on purpose; hitting one by accident is most unlikely, indeed.
All that being said, it is reasonable to suppose that the Solar System's asteroid belt has undergone thinning since it first formed. It was likely denser (but still rarefied) in the early epochs of the Solar System, and events and effects such as migration of the giant planets (resulting either in captures as irregular moons or complete ejection from the Solar System) and rare-but-catastrophic collisions have doubtless reduced the population somewhat.
Thus, one might be able to make an argument for an asteroid belt denser than that of the Solar System, but the Hollywood depiction is still more suited to planetary rings.