Explosion of an asteroid in orbit

Question

I've learned from an answer here on this site that if a body were to split apart in orbit the center of mass will continue to be on the same orbit. (Couldn't find the post)

But let's say an asteroid is to blow up in two pieces such that the smaller piece reverses its direction and velocity and thus stays in the same orbit, but going backwards.

But now the bigger piece must be faster than the velocity of the center of mass, this higher velocity changes the orbit of the bigger mass.

Now the center of mass is not in line the smaller piece continues the same orbit while the bigger piece changes the orbit. This doesn't seem right. What went wrong?

For example this question https://www.toppr.com/ask/question/an-asteroid-orbiting-around-a-planet-in-circular-orbit-suddenly-explodesinto-two-fragments-in-mass/

An internal explosion causes the center of mass to move farther and farther away from the Earth.

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I'm very certain I've made astronomical blunders but I would like to know my mistake.

Answer 1

if a body were to split apart in orbit the center of mass will continue to be on the same orbit.

This is not always true. It would be true if we were talking about a projectile in a uniform gravitational field, but for an orbiting satellite the gravitational field is non-uniform. That means that for large displacements from the center of mass the split portions may experience very different gravitational fields than they would have experienced on the original orbit. The center of gravity is not the same as the center of mass for a non uniform gravitational field.

Answer 2

If a body splits in orbit, the center of mass of the body won't necessarily be in the same orbit, before and after, even though the sum of angular momentum of the pieces will be conserved. As an example, consider an asteroid broken into 2 pieces which both have escape velocity in opposite directions. The center of mass of that asteroid wouldn't even have a bound orbit with the main body anymore.