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The question What's the (particle) density of the asteroid belt? is about the density of objects in the asteroid belt.
As a follow up related question and what I am interested in:
When we send spacecraft through the asteroid belt how much finger crossing is going on? Is it better to travel over the solar system rather than through it?
The asteroid belt isn't nearly as dense as popular media makes it out to be.
An answer from the Dawn Mission's FAQ, specifically "What is the average distance between individual asteroids? (6/13/10)", helps here.
Asteroids are not distributed uniformly in the asteroid belt, but could be approximated to be evenly spaced in a region from 2.2 AU (1 AU is 93 million miles, or the average distance between Earth and the Sun) to 3.2 AU from the Sun and extending 0.5 AU above and below the ecliptic (the plane of Earth's orbit, which is a convenient reference for the solar system). That yields a volume of roughly 16 cubic AU, or about 13 trillion trillion cubic miles. (Note: space is big!)
If there were 2 million asteroids 1 mile or larger in that volume, each asteroid would have 6.7 million trillion cubic miles to itself, so the average distance between individual asteroids 1 mile in diameter or larger would be about 1.9 million miles. That is nearly 8 times the distance between Earth and the Moon.
Just for something interesting, it looks like your gif only covers the Jupiter trojans, which are the asteroids that Jupiter shepherds around the solar system at its Lagrange points. A more accurate image is available on Wikipedia.
The asteroid belt is roughly 6 Astronomical Units wide, and so when it is drawn only 600 pixels wide with each asteroid a handful of pixels wide, you end up with each asteroid being five times bigger than the Sun!
I've borrowed a small piece of the image used in @jos' excellent answer to show what I mean.
As pointed out in this answer, Wikipedia says:
Contrary to popular imagery, the asteroid belt is mostly empty. The asteroids are spread over such a large volume that it would be improbable to reach an asteroid without aiming carefully.
The Exposing PseudoAstronomy blogpost Asteroid Belts – Proximity of Rocks and Why Navigation Is Not Dangerous (Sorry, Han Solo) says:
How Many Asteroids of What Size?
In terms of what is known, there are about 20,000 asteroids between 2-3 km, which is about the smallest that we likely have a complete sampling of. What that statement means is that, while we have identified asteroids that are smaller, our detection technology is not good enough to have found all of the asteroids that are smaller.
If we extrapolate, assuming a -3 power low, down to, say, 100-meter asteroids, there are probably ~82 million asteroids that are ~100-200-meters across. If we extrapolate further, down to 1-meter asteroids, then we really have a gargantuan number of objects – about 1014 (100 quadrillion) objects of that size. That’s quite a lot.
What Does this Mean for Navigation?
If we add up all of those objects, we have about 1.2×1014 asteroids larger than 1 meter. Now, let’s look at the asteroid belt. It stretches from 1.8 to 3.3 A.U., which is a distance of 1.5 A.U., or about 225,000,000 km. That’s a fairly large distance (that’s actually about the distance between the Sun and Mars).
The area of a disk that size, however, is gargantuan: A = π · r2 = π · ((3.3 A.U.)2 – (1.8 A.U.)2) = π · (1.7·1017 km2) = 5.4·1017 km2. That is a huge area. Simple division shows that each asteroid, regardless of its own size, could have 4,500 km2 all to itself – a little bit more than the entire U.S. state of Rhode Island.
And that’s if they were all just in one plane. In reality, they occupy a volume of space, some orbiting “above” or “below” others (where those terms are relative to the plane that the Earth’s orbit makes).
Even if we cut the size of asteroids in half again, and were interested in all asteroids larger than half a meter (1.5 ft) in size, then we have 8 times as many asteroids, but each one still has over 500 km2 all to itself, and even more space if we consider the vertical component.
What does this mean for navigation? It’s easy! In fact, you really have to try to hit an asteroid, at least in our own belt. And so, the next time you see a tiny ship careening through an asteroid field in a TV show or movie, remember that in real life, asteroid belts really aren’t that dangerous for navigation.
@AlanSE's answer to the related but different question What's the (particle) density of the asteroid belt? shows an illustration of the power law behavior of size at large, observable scale. Unfortunately the link to the source (dated 2001) is now broken.
