Imagine a binary planet where one of the planet is of the size and mass of Jupiter (planet A) while the other would be like the Earth's moon (planet B). Imagine that this system is extremely stable (lasts billions or many more order of magnitude of time) and is put inside a constant flux of small to medium sized asteroids (not big enough to break the planets apart).
I'm wondering whether planet B would benefit from the bigger's planet gravitational field in such a way that in the long run it would tend to have the same size/mass than planet A.
Or whether this is wrong and planet A would always have a huge disparity regarding the mass and size compared to planet B.
Here are some thoughts: Compared to being alone, originally at least, planet A is being shielded from the asteroids because of planet B. While for planet B, compared to being alone under the flux of asteroids, there is a huge benefit of getting hit by asteroids because of planet A's gravitational field. And so at first it seems to me that the rate of being hit would favor planet B compared to planet A. By rate I mean "x asteroids per y units of mass".
The problem is likely solvable by computational simulation but I'm not skilled enough to do it.
I'd like to hear your opinions on the question.
Edit: For this problem we'll neglect relativistic effects. To make things simpler we can consider the densities of the planets and asteroids to be the same and that the shape of both planets is always spherical.