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Rock Distribution

Rock Distribution

Building Planets – [nasa.gov]

A beautiful artistic illustration of an asteroid belt (link above) shows a common misconception of asteroid spacing that is prevalent in both science fiction and the public mind. We love to watch Hans Solo or some other pilot skilfully guide a spaceship through the clutter of an asteroid field. But in reality (at least in our solar system) our asteroid belt is nothing like that. The rocks are widely scattered and there is little chance of colliding with one.

In order to come up with an estimation of how spread out the rocks are, we will need two numbers:

  1. The number of rocks in the asteroid belt
  2. The amount of space across which they are distributed

Here is a good starting point:
Asteroid Belts – Proximity of Rocks and Why Navigation Is Not Dangerous (Sorry, Han Solo) – [pseudoastro.wordpress.com]

In general, the distribution of sizes follows what’s known as a “power law” distribution, where the number of small asteroids grows much more quickly than the reduction in size. The slope of this power law is generally estimated to be -3. What that means is that every time you halve the size of an asteroid, you have 8 times as many. So say there are 100 10-km asteroids. With a -3 power-law slope, that would mean there are 800 5-km asteroids. And 6400 2.5-km asteroids. But only ~13 20-km asteroids.

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.

This gives us at least a working starting point for estimating the number of rocks (until somebody else offers a more accurate methodology). Using the numbers represented above and adjusting to find the number of rocks for a 3 meter minimum size (about the size of an average bedroom – 10ft cubed), we come up with an estimate that there might be around 2.7 trillion rocks of that size or greater in the asteroid belt.

[Set up your own spreadsheet and you can play with the numbers all you like]

Next we need to calculate the area (in sq kilometers – flat space) that the belt occupies. Data obtained from the “Asteroids – Dynamic Site” at: AstDyS – Proper Elements and the following mapping is useful:

Asteroid orbital distance vs inclination

Asteroid orbital distance vs inclination

Across the bottom axis is shown the orbital distance from the Sun in AUs (Astronomical Unit) where “1.0” represents the orbital distance of the Earth = 149,600,000 kilometers. Closing the edges of the data set in to 2.1 AUs and 3.3 AUs defines the main body of the asteroid belt. Calculating the areas of both the inner edge disk and the outer edge disk and subtracting them from each other gives an estimated area for the main asteroid belt of 4.56 x 10-17th square kilometers.

Dividing that area estimation (flat space) of the main belt by the estimated number of 3-meter rocks and you get a result of a little over 169,000 square kilometers of space reserved for each rock. But that is only using flat space for the calculations and we know the asteroid belt is distributed in three dimensions.

The ecliptic plane is the flat plane that contains most of the orbits of the planets. The asteroid belt is centered around the ecliptic plane but also extends out from it, both above and below. In order to determine how spread out from the ecliptic plane the rocks are, the angle of inclination of the orbits is needed. The data from the image above shows the inclination of asteroid orbits in degrees. At the inner edge of the belt, the inclination is around 7 degrees and at the outer edge it grows to around 24 degrees. By using the tangent function and the orbital distance, it is possible to calculate the distance involved at the opposite side of a right triangle from the angle of inclination. Doubling that (to give both the distance above and below the ecliptic plane) shows the depth of the asteroid belt to be about .52 AUs (or 77 million kilometers) at the inner edge and 2.94 AUs (or 440 million kilometers) at the outer edge. An average depth can be estimated at 1.65 AUs or 247 million kilometers and multiplied by the area of the disk to give a total volume estimate of 1.35 x 10-26th cubic kilometers of space occupied by the asteroid belt!

[note – this is a huge volume of space that completely dwarfs every other feature of our Solar System – the asteroid belt at it’s outer edge that is approximately 3 AUs from the Sun is almost 3 AUs wide at that point]

Divide that big number by the estimated 2.7 trillion rocks of 3-meter or greater size and each rock still has 4.18 x 10-14th cubic kilometers of space or an empty sphere with a diameter of around 93,000 kilometers.

A lot of these numbers are estimates and we simply don’t know how many rocks there are yet or exactly where they are. But it’s obvious that they are widely spread out and for the most part will not pose a navigational risk of any great worry.

SEE ALSO:
The Rocks
Belt Geography

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