# Hypothetical binary planet of different size and mass, would they tend to have the same size and mass?

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.

Intuitively, I would expect the bodies to become more disproportionate over time. The more massive one has a larger Hill sphere, which makes it more likely to capture passing asteroids, planetoids, etc. Since the only way for a celestial body to gain mass is to capture passing bodies, this does not favor the sizes evening out.

• I agree that planet A has a larger Hill sphere but in my case planet B is entirely inside planet A's Hill sphere. So I am not sure whether only knowing about the size of the Hill sphere is enough to answer the question. On the other hand I've talked to a friend about this question and he helped me to answer it involving cross sections (a topic I never studied). I would like to run some simulations and then write my answer to the question here. Meanwhile if you have any comment feel free to post. – thermomagnetic condensed boson Dec 14 '15 at 12:31

Here's my answer after having being helped a lot by a friend. Considering a scattering problem (something I have never studied before, neither in CM nor QM) in the x-y plane. I assumed that asteroids are "fired" initially in the x-direction with speed $v_0$ and that they are thrown one by one. This makes the magnitude of the angular momentum to be $L=myv_0$. The effective potential of the system planet-asteroid is $U_\text{eff}=U(r)+\frac{L^2}{2mR^2}$ where $U(r)$ is the Newtonian gravitational potential. I call $f(r)=U_\text{eff}-E_0$ where $E_0$ is the initial energy of the asteroid (worth $\frac{mv_0^2}{2}$). Now I ask $f(R)=0$ (where R is the radius of the planet; this implies a collision), this yields $y^2=R^2+\frac{2GMR}{v_0^2}=R^2+\frac{8\pi G\rho R}{3v_0^2}$. This is the radius squared of the cross section for a planet of radius R (and uniform density rho). Now the quotient between the cross section radii of the 2 planets gives me the probability for a planet to be hit. For instance if the quotient gives me 1/3 then there is 25% chance a planet will be hit and 75% chance the other will be hit, assuming a collision occurs with probability 1. Here's the python code that I used to run the simulations.

The result is that in the end the smallest planet's radius increases at a faster rate than the bigger's planet radius (as my first guess). And this is valid no matter the size of the planets I've tried (quite a few). Thus in the limit or maybe after some finite number of asteroid hits, both planets have the same size/mass; or at the very least, tend toward this situation. So my answer to the original question is a yes. I'd appreciate any comment and/or answer.

Edit: The ratio of the radii tends to 1 is what I mean I've found out with my model and programming code.