Concerning asteroids and a trojan habitable planet

Question

So I found a question on this site that talked about the possibility of having a habitable planet in a stable trojan position (a planet located at Lagrangian point L4 or L5 of a much bigger object). And I thought it was cool, and decided to make it the homeworld of a spacefaring civilisation of my worldbuilding project.

So I decided to create a planetary system consisting of a central sun-sized star (1.1 solar masses), an orbiting brown dwarf (20 Jupiter masses, distance to the star: 1.45au, quasi-circular orbit), and a small rocky planet of half the Earth's mass at L5 point of the brown dwarf. That's not all but the rest of the system isn't relevant.

Then I started thinking about the planet's details and a few questions came to my mind. One of which I cannot answer at all is the following :

How would asteroids and comets be affected ? The brown dwarf would attract them all even more than Jupiter. Would a significant amount of them (or their debris) be redirected towards the planet, thus increasing the overall amount of asteroids hitting the planet? Or would the dwarf attract all the asteroids coming to the planet, completely protecting it ? Overall, would more or less asteroids collide with the planet ?

My knowledge is limoted on how exactly Jupiter affects asteroids, other that it attracts them and thus protects the rocky planets. In this situation the Jupiter-like object is much more massive, and my planet is much closer to it, and more or less comets and asteroids could affect the habitability and amount of natural resources of my planet, especially by adding water. On the other side, if asteroids regularly hit the planet, it might make it a harsh environment, that would definitely affect the native's culture. Hence my question.

Answer 1

Well, the full answer to your question is not really possible as mankind doesn't fully understand Jupiters influence on comets and asteroids until now. The newest study regarding the 'shielding by Jupiter'-theory I could find is this one and I will try to answer your question mainly on its conclusions.

One of the most important core-facts the study accomplished by simulating the movement of more than 100k objects in a period of 10 million years under the influence of different Jupiter-masses was the following:

As the mass of Jupiter is increased, the impact rate from objects moving inwards from the Edgeworth-Kuiper belt first rises to a peak, then falls away again. The end result is that a Jupiter like our own (one Jupiter mass) provides an almost equivalent amount of shielding to no Jupiter at all! More importantly, were our Jupiter smaller, then the impact rate from this Centaur-derived population of objects would have been higher than in either extreme case. In the worst case, with a Jupiter around 0.2 times the mass of our planet, the impact rate would have been significantly higher than at either extreme!

The study used the actual mass of Jupiter as a max for their simulation but the statistics suggest that a significantly higher mass for Jupiter (as your brown dwarf has) really could have a shielding effect for objects in the same or a lower orbit. I conclude that on their following statement:

When there is no Jupiter, then very few objects acquire Earth-crossing orbits (Saturn is much less effective than Jupiter), and so the impact rate is particularly low. As the mass of Jupiter increases, the efficiency with which it places objects onto Earth-crossing orbits also increases, and so the impact rate rises. Eventually, though, as the mass increases still further, Jupiter becomes ever more efficient at ejecting the particles from the solar system entirely, and removes them from Earth-crossing orbits on ever shorter timescales. Thus, even though more objects are flung inwards, they are removed so rapidly as to pose significantly less of a threat. The shorter their residence in the inner solar system, the fewer opportunities they have to hit the Earth, and the lower the risk they pose.

The biggest difference between your system and the study is, that your planet is in the same orbit as your brown dwarf and the study is for planets in a lower orbit than Jupiter, but by the concept of the simulations they made I think it could get similar results for your planet. At least I am sure the planet at your Langrange-Point L5 should not get a higher impact-rate than a planet on a lower orbit around the sun of your star.

Long story short: I think a longtime-habitable planet at L5 within your constellation is plausible.