Is there any function that can describe the distribution* of asteroid diameters expected within an*' asteroid belt?
*: distribution as frequency of occurrence within the asteroid belt or absolute numbers, not as positional distribution or orbits inside the belt.
*':Not THE asteroid belt. So the answer could/should fit Sol's 'main' asteroid belt, but also any rather usual one on any other star system.
Possible independent variables used by the model given
The fewer used the better, as the model will be broader, but if by introducing any the precision of the result increases by a magnitude, then you are welcome. The variables at my disposal to find out the distribution of sizes are:
- Initial mass of the belt on system formation.
- Inner and outer radius of the ring.
- Current age of the system.
- Star(s) radius, mass, luminosity and temperature.
- Planets around, and their masses and orbital characteristics. (this would not include planetesimals produced by the model, of course)
You can add any constant that you find necessary.
The more general the solution presented, the better, but any amount of them taken are acceptable to answer the question.
- Considering a star system old enough to be stable.
- Constant density of the bodies in it.
- It's a rather standard main belt. This means that it does not need to represent oort cloud or anything further than ~100AU, neither heavily disturbed belts like Kuiper or singular through expectedly common ones like trojan 'belts'. (but will be very welcome if does!)
Other considerations and previous research
The model needs to fit only standard relatively usual scenarios: there happen to be the conditions that allow the formation of and asteroid belt but prevent it's accretion into a planet, (e.g., a Jovian around) but not any other strange condition (like gas planets drifting inwards in the formation period and altering significantly the distribution).
To be precise, I'd like to find out the most general of such expected distributions. If it's relevant o known, I'll be happy to have the distinct versions for the 'rocky' zone of the system, without volatiles, and the icy part.
I found an article about the distribution of asteroids in our main asteroid belt, but couldn’t get anything like what I ask from it: http://orbit.psi.edu/~tricaric/pdf/skads.pdf
Wikipedia also has something to say about it in the case of the Kuiper Belt: https://en.wikipedia.org/wiki/Kuiper_belt#Mass_and_size_distribution But as there seem to be so many things that do not fit the expected models about that belt, I don't dare make generalizations from it.
I understand that there won't be probably any accepted model of the kind I'm looking for, and giving an exact answer might involve doing original research, but I'd like to find out at least an acceptable and credible rule of thumb that has some decent foundation.
It's allowed that the diameter of the given asteroids is up to be enough to be considered planetoids (around 10^3 km), but if it's significantly bigger, please tell me how it does not clean it's orbit or form a 'trojan' type of ring instead of a typical more or less evenly spreaded ring, and I'll be happy to consider correct the 'magno-planetoids' :D.
Edit 1: Formatting changes and better parametrization of what's expected to be in and what may be left out of the answer.
Edit 2: I've found some interesting article: https://www-n.oca.eu/morby/papers/fossileSFD.pdf I'll post what I find in it as an answer if I'm able to get it from there, though it looks like it will only model our belt, and so not be enough.