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Two masses, $m$ and $2m$, orbit around their CM (center of mass). If the orbits are circular, they don’t intersect. But if they are very elliptical, they do. What is the smallest value of the eccentricity for which they intersect?

My doubt is: If their trajectory intersect then it intersects at CM which means the two particle crashing up in the CM making the force between them tending to infinity. How's this possible? Moreover the differential equation for the relative separation of the two particles is coming up as: $r'' = \frac{3 G m}{r}$ . Where $r$ is the relative separation.

So how do we conclude that the trajectory of the particle is a circle or ellipse?

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The two ellipses can intersect but they don't intersect at the center of mass. (How could they when the center of mass is at a focus of each ellipse? The focus of an ellipse is not on the ellipse but rather inside it.) The particles never collide (unless the eccentricity is 1) because they are always on opposite sides of the center of mass.

I find that when the mass ratio is 2, the minimum eccentricity for intersection is 1/3. (I'll leave that calculation to you.) The orbits look like this:

enter image description here

The center of mass is at the origin.

For smaller eccentricity (here 0.2), they don't intersect:

enter image description here

For larger eccentricity (here 0.4), they intersect at two points:

enter image description here

Your differential equation has three different errors. It should be a vector equation; the right-hand side should be negative; the right-hand side has the wrong power of $r$.

The correct equation is

$$\frac{m_1m_2}{m_1+m_2}\frac{d^2\vec{r}}{dt^2}=-\frac{Gm_1m_2\vec{r}}{|\vec{r}|^3}$$

so you got the factor of 3 correct.

How do we conclude that the trajectory is a circle or ellipse (or a line segment)? By solving the correct equation to get $r(\theta)$; these are the only solutions that don't go off to infinite $r$. Whether it is a circle or an ellipse depends on the initial conditions.

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My doubt is: If their trajectory intersect then it intersects at CM which means the two particle crashing up in the CM making the force between them tending to infinity. How's this possible?

Orbits intersecting is different than the masses colliding.

The orbit is the set of spatial points covered by the trajectory. i.e. orbits don't contain any temporal information about the orbiting masses.

Saying two orbits intersect just means that they share at least one point in space. However the masses can occupy these points in space at different times.

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