How long until the planets collide?

[jgens]

Homework Statement

Two planets of masses $m_1$ and $m_2$ with radii $r_1$ and $r_2$ respectively are orbiting their common center of mass at some initial distance $x_0$ from each other with angular velocity $\omega_0$. Find the amount of time it takes for these planets to collide.

Homework Equations

N/A

The Attempt at a Solution

So far, I've drawn my free-body diagram with the axes in the Center of Mass reference frame, figured that the position of the center of mass remains constant since no net external force acts, and that conservation of momentum applies since no net external torque acts either. However, I'm having difficulty setting the problem up (constructing a series of differential equations to solve) and would appreciate any advice.

Thanks!

 

[jgens]

Bump (this was getting towards the bottom of the page).

 

[jgens]

Bump (getting towards the bottom of the page again).

 

[JANm]

Find the amount of time it takes for these planets to collide.
Thanks!

Hello jgens
In the way you put it it is a potential equation. For colliding you need something diffusive. Rotational energy diminishing because of friction or any other diffusion. For instance if a speaker moves particles a comoving for some time, that is called sound. Since this comovement diffuses the sound deminishes and the particles are moving statistically erradic again.
greetings Janm