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I'm putting together a solar system for a story I'm working on, and in this solar system the most important planet is a particularly large Ice giant and its big moon nestled in the star's habitable zone (the sun is basically identical to ours).

I wanted to ask about Trojan Asteroids, I'm trying to understand what factors may influence a large pile up of big Asteroids at this planet's L4 and L5 lagrange points.

I understand that Jupiter has the most studied Trojan systems, some of the objects are quite large, in excess of 100 kilometers in diameter. Wikipedia tells me that there might be as many as a million Jupiter Trojans larger than one kilometer. Additionally Neptune probably has even more, and larger ones too. However Saturn and Uranus seem to have less than the other giant planets. Is Trojan density thus directly correlated with proximity to belts of small bodies, like the Asteroid and Kuiper belts? Is it affected by the size of the dominating body, or proximity to the sun? Would my Neptune like planet in the same position as earth be likely to have a significant collection of fairly large Trojans(about the size of Gaspra say), with or without an asteroid belt nearby?

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Much of the density of materials in the Solar System has to do with the movement of the giant planets during the early history.

While the planets were forming, they were also plowing through dense clouds of dust, gas and primordial materials (cometary nuclei and protoasteroids). Much of the material was incorporated into the growing planets, but some was far enough away to be either accelerated by the forming planet's gravity (flinging it to farther out orbits or even solar escape velocity) and slowing the planet down by a corresponding amount (given the huge disparity in mass, the small particles were shot out at high velocity while the planet slows down by an correspondingly tiny amount).

This cleared much of the inner Solar System and pushed a lot of material out into the Kuiper belt.

The density of the remaining material was set by the resonances between various planetary orbits. Materials which fell into the resonant gaps could remain there, while items outside would be "pumped up" by repeated gravitational pulls from Jupiter etc. and fall sunward or outwards. These materials would, over the course of eons, eventually settle into other places of relative gravitational stability, such as the Jovian L4 and L5 points.

Materials in the outer system would have been affected to a lesser extent since the outer planets are much smaller and farther apart (giving them less gravitational influence), not to mention the long orbital periods, which mean the resonant zones would be affected far more slowly. A single gravitational "kick" from Jupiter every 12 years probably has much more influence than a tiny nudge from Uranus every 84 years, leaving much more open space and lower material density than might otherwise be the case.

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