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This is similar to a previously asked question, but I am just asking about theory rather than observational evidence. Assuming there were a much larger number of protoplanets in the early Solar System, what percentage of these would theoretically have been ejected (orbital velocity increased beyond escape velocity due to interaction) and what percentage would have been just moved out to a more distant orbit (say, in the Kuiper Belt or the Oort Cloud)? In other words, should we expect to find rocky worlds in the far outer Solar System that were formed inside the current radius of the asteroid belt?

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  • $\begingroup$ note: I did a quick calculation and at the Earth's distance from the sun, the escape velocity would be about 42 km/sec (the Earth's orbital speed is about 30 km/sec). $\endgroup$ – Jack R. Woods Nov 26 '15 at 11:36
  • $\begingroup$ qualification: in previous comment, this would be vertical escape velocity (ie. radial direction away from the sun) $\endgroup$ – Jack R. Woods Nov 30 '15 at 22:58
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We can identify two distinct realms of planetesimal formation - the inner Solar System and the outer Solar System. The initial group of small bodies1 in the inner section of the protoplanetary disk was rather quickly accreted into planets via various interactions; some bodies grew bigger and thus dominated the disk around them. In the outer parts of the disk, many small bodies were accreted into the cores of the giant planets. However, not all were accreted, and there may in fact have been a second generation of planetesimals that formed via gravitational collapse due to disk instabilities.

We can compute timescales for this "cleanup" phase, the period during which small bodies were either accreted onto protoplanets or ejected to larger orbits or out of the Solar System entirely. In the inner section of the disk, among the terrestrial planets, it was on the order of $\sim10^8$ years, while in the outer section, it was on the order of $\sim10^9$ years. However, the giant planets may actually have finished accreting matter before the terrestrial planets.

This means that the small bodies that were ejected or boosted to larger orbits were largely from the outer Solar System, terrestrial planetary migration effects aside2. Various N-body simulations agree that a total of $\sim300M_{\oplus}$ of small bodies was ejected from the outer disk, due to interactions with Jupiter and the formation and evolution of Uranus and Neptune. We know that the Oort Cloud's mass is on the order of $\sim1$ to $\sim10M_{\oplus}$, and the Kuiper Belt and scattered disk have masses an order of magnitude or two lower. Therefore, only a small fraction - certainly less than 10% - of the small bodies could have been ejected into the Oort Cloud.

If you want to visualize some of this, check out this video from a simulation of the Nice Model3. It shows the orbits of the four giant planets (Jupiter in red, Saturn in orange, Uranus in purple, and Neptune in blue). It shows Neptune and Uranus swap orbits, and also shows the dispersal of a large group of small bodies outside the original orbit of Uranus (which began further out than Neptune). The video shows the simulation over the course of a bit more than one billion years, and you can see a drastic clearing out of the extreme outer disk over this timescale.

The above results came before the development of the Nice Model and its variants, as well as the possible discovery of Planet Nine and the discoveries of many small objects in the outer Solar System, beyond Neptune. They should therefore not be taken as totally accurate descriptions of the formation of the Solar System - the video in particular is likely incorrect - and I'm guessing I'll get some flak for using them as the basis of this answer.

Obviously, many N-body simulations of the Solar System have been done, and over the last decade, a substantial number use some variant of the Nice Model. Some use planetesimal disks - excluding the masses of the eight planets - on the order of $\sim10$ to $\sim10^2M_{\oplus}$; Nesvorny & Morbidelli (2012) (two of the architects of the model) found a sweet spot, so to speak, of $20$ to $50M_{\oplus}$, depending on the exact result. As I understand it, masses of the surplus disk around these values fit the Nice Model well, and so perhaps we should drop the total mass of the ejected small bodies by a factor or about ten.

Footnotes

1 I used this term quite a lot here; it refers to bodies with masses in the planetesimal regime and lower. The reason for this is that a substantial amount of the ejection came after the inner planets had formed, and the remaining bodies were small.

2 This should not be taken to mean that no or very little matter from the inner Solar System was ejected, especially if we assume some movement among the orbits of the inner planets (see Meech et al. (2016)).

3 Originally brought to my attention through a question on Physics Stack Exchange by Kyle Oman.

References

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  • $\begingroup$ What about our superearth(s)? Isn't our system quite atypical in not having one and instead having a very small first planet? $\endgroup$ – kubanczyk Oct 28 '16 at 21:28
  • $\begingroup$ @kubanczyk Are you sure you don't mean Hot Jupiters? The fact that a lot of exoplanetary systems have them is detection bias; they're extremely easy to find, compared with most other planets. $\endgroup$ – HDE 226868 Oct 28 '16 at 21:52

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