Probability of exoplanets getting ejected by supernova

Question

The large population of rogue planetary mass objects in our galaxy is often stated to have stemmed from gravitational ejection from a young system of from the collapse of small gas clouds. But I've been wondering, could supernova explosions give rise to a small percentage of these objects? So far we have discovered exoplanets/brown dwarfs around stars up to approximately 10 solar masses, which is past the threshold for a supernova. Would/do jovian mass exoplanets/low mass brown dwarfs on long period orbits survive the resulting supernova, and would the outward strength of the explosion + the mass loss of the parent object be enough to eject the body into interstellar space? How common would this event be? And what would remain of the body? The exposed core of a jovian mass gas giant or a large planetary object from a brown dwarf? Or something entirely different?

Answer 1

If you have "non-conservative mass loss", where the mass expelled from one object in an orbit is not wholly accreted by the other, then the system can become unbound.

For the simple case of an originally bound, circular orbit, the total energy is $-GM_1M_2/2a$, where the kinetic energy component is $GM_1M_2/2a$ and a potential energy component of $-GM_1M_2/a$.

If object 1, say a high mass star, quickly loses more than half its mass in a supernova, and that mass is not accreted by the other object, then while the kinetic energy is unchanged, the magnitude of the potential energy is more than halved, the total energy exceeds zero, and the system becomes unbound.

So yes, if high mass stars have planets, and the high mass star explodes as a supernova, then because it is usually the case that more than half the stellar mass escapes the system, then the planets would become unbound.

Such events are unlikely to produce many "rogue planets", as a fraction, simply because high mass stars are very rare compared to say solar mass stars. There are other ways to eject planets during close encounters of these more common types of stars, especially in their dense birth environments.