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How pristine and ancient are asteroids really? I suppose they melt and reform when they collide with each other. How frequently do they collide with each other?

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How ancient they are, I'd have to say as old as the solar system. Collision frequency:
Collision amongst Asteroid Belts From ref 1 below: Based on the particle in a box model which assumes a homogeneous distribution in the interactive volume, the intrinsic collision probability is computed to be around $5 \times 10^{-18} km^{-2}yr^{-1}$.
However Wetherill(ref below) have some modifications to Opik's formulation for collision between two elliptical orbits, suggests that the impact is nearly independent of the orbital elements of the test body and is nearly half of the value obatined from PIAB model. It also seems to suggest that intrafamily collisions are higher than collision amongst non family objects.

Collision with the Earth Here it characterized in various classes:

  1. bolides or upper atmospheric explosion,
  2. land impacts leading to local or regional devastation,
  3. ocean impacts leading to regional disasters,
  4. global ecological catastrophe,
  5. mass extinctions

Based on the scale, the impact events are categorized as harmless, Class-1, to devastating, Class-5.

The 440-KT Chelyabinsk event of 2013 belongs to impact hazard class 2, causing regional or local devastation. This was caused by a mere 20 meters wide meteor entering the atmosphere of Earth. For a cumulative number of impacts described for a given reference time interval of $T_{ref}$ = 1Ga ($10^9$ years) which is long enough to statistically include a meaningful number of impacts from all the classesabove, the class 2 impacts cumulative will be $2\times 10^6$ , with small bodies falling in the size range of 20-100 m. Where the characteristic time scale is of $10^2$ a.

For class 3-5 events leading to regional disasters to global extinction events, the cumulative number of impacts within the reference interval is $10^4$ with characteristic time scale being $10^5$ a. These events can lead to large devastation ranging from an entire country or region (class 3, with an explosive energy of approximately 104 MT TNT), to mass extinctions (class 5 releasing an approximate energy 108 MT of TNT). An example from the history for such event is the classic K-T extinction events 65 Ma ago that wiped out more than 99% of the living species.

References:
1. http://articles.adsabs.harvard.edu//full/1992LPI....23..283D/0000283.000.html
2. http://www.springer.com/fr/book/9780792354666
3. http://onlinelibrary.wiley.com/doi/10.1029/JZ072i009p02429/abstract
(PS: If anyone has a link to the open access for final paper, or a pdf; I would really appreciate a copy.)

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    $\begingroup$ I might be wrong, and the frequency of collisions seems like an extremely vague question, but I fail to see how the asteroids' impact hazard category has anything to do with how often the asteroids collide? $\endgroup$ – V-J Jan 4 '16 at 9:22
  • $\begingroup$ It does actually, the number of collisions is dependent on the size of the small body. I will put a reference. Besides it's easily observed with the number of larger impacts we see every 100 years or so, for eg Tunguska event by a large asteroid took place almost a century ago, while a mass extinction event took place over million years ago and is assumed to be overdue now. $\endgroup$ – Astroynamicist Jan 4 '16 at 11:53
  • $\begingroup$ Good to know about the source, and I remember most of your points from my solar system physics courses, but still the question is about asteroid-asteroid collisions. Exactly finding out the frequency would be a pestering multi-body perturbed orbital equation to simulate, but it would be interesting to know if anyone has worked any sort of figure for the frequency of collisions. $\endgroup$ – V-J Jan 4 '16 at 12:03
  • $\begingroup$ Yeah, I get it, but, the original question did not specified asteroid-asteroid collision. I'll look into it and see what I can find. But you're right it will involve simulating multi-body perturbation model, but there must be analytical expressions obtained from statistical analysis. And I think if we assume simple nodal interactions between each orbit then we can use Wetherill modification, to particle in a box model. But since most asteroids lie along the same place there would be some corrections to it. Also, how do you edit a comment through the app? $\endgroup$ – Astroynamicist Jan 4 '16 at 12:43
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    $\begingroup$ Apparently the comments cannot be modified even thorugh webpage. On the bright side, somebody has done the computations for us. Check the updated answer. $\endgroup$ – Astroynamicist Jan 5 '16 at 1:10

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